LMFDB - Embedded newform 429.2.i.a.133.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [429,2,Mod(100,429)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(429, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("429.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 429 = 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	429.i (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:	$3.42558224671$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			133.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	429.133
Dual form			429.2.i.a.100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +2.00000 q^{5} +(-0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +2.00000 q^{5} +(-0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(0.500000 + 0.866025i) q^{11} +2.00000 q^{12} +(3.50000 + 0.866025i) q^{13} +(1.00000 + 1.73205i) q^{15} +(-2.00000 - 3.46410i) q^{16} +(-1.00000 + 1.73205i) q^{17} +(2.00000 - 3.46410i) q^{19} +(2.00000 - 3.46410i) q^{20} -1.00000 q^{21} +(-2.00000 - 3.46410i) q^{23} -1.00000 q^{25} -1.00000 q^{27} +(1.00000 + 1.73205i) q^{28} +(4.00000 + 6.92820i) q^{29} -3.00000 q^{31} +(-0.500000 + 0.866025i) q^{33} +(-1.00000 + 1.73205i) q^{35} +(1.00000 + 1.73205i) q^{36} +(-5.00000 - 8.66025i) q^{37} +(1.00000 + 3.46410i) q^{39} +(4.00000 + 6.92820i) q^{41} +(3.50000 - 6.06218i) q^{43} +2.00000 q^{44} +(-1.00000 + 1.73205i) q^{45} +8.00000 q^{47} +(2.00000 - 3.46410i) q^{48} +(3.00000 + 5.19615i) q^{49} -2.00000 q^{51} +(5.00000 - 5.19615i) q^{52} -10.0000 q^{53} +(1.00000 + 1.73205i) q^{55} +4.00000 q^{57} +(-4.00000 + 6.92820i) q^{59} +4.00000 q^{60} +(-6.50000 + 11.2583i) q^{61} +(-0.500000 - 0.866025i) q^{63} -8.00000 q^{64} +(7.00000 + 1.73205i) q^{65} +(-5.50000 - 9.52628i) q^{67} +(2.00000 + 3.46410i) q^{68} +(2.00000 - 3.46410i) q^{69} +(3.00000 - 5.19615i) q^{71} -13.0000 q^{73} +(-0.500000 - 0.866025i) q^{75} +(-4.00000 - 6.92820i) q^{76} -1.00000 q^{77} -7.00000 q^{79} +(-4.00000 - 6.92820i) q^{80} +(-0.500000 - 0.866025i) q^{81} -6.00000 q^{83} +(-1.00000 + 1.73205i) q^{84} +(-2.00000 + 3.46410i) q^{85} +(-4.00000 + 6.92820i) q^{87} +(-4.00000 - 6.92820i) q^{89} +(-2.50000 + 2.59808i) q^{91} -8.00000 q^{92} +(-1.50000 - 2.59808i) q^{93} +(4.00000 - 6.92820i) q^{95} +(3.50000 - 6.06218i) q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{4} + 4 q^{5} - q^{7} - q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^4 + 4 * q^5 - q^7 - q^9	$ 2 q + q^{3} + 2 q^{4} + 4 q^{5} - q^{7} - q^{9} + q^{11} + 4 q^{12} + 7 q^{13} + 2 q^{15} - 4 q^{16} - 2 q^{17} + 4 q^{19} + 4 q^{20} - 2 q^{21} - 4 q^{23} - 2 q^{25} - 2 q^{27} + 2 q^{28} + 8 q^{29} - 6 q^{31} - q^{33} - 2 q^{35} + 2 q^{36} - 10 q^{37} + 2 q^{39} + 8 q^{41} + 7 q^{43} + 4 q^{44} - 2 q^{45} + 16 q^{47} + 4 q^{48} + 6 q^{49} - 4 q^{51} + 10 q^{52} - 20 q^{53} + 2 q^{55} + 8 q^{57} - 8 q^{59} + 8 q^{60} - 13 q^{61} - q^{63} - 16 q^{64} + 14 q^{65} - 11 q^{67} + 4 q^{68} + 4 q^{69} + 6 q^{71} - 26 q^{73} - q^{75} - 8 q^{76} - 2 q^{77} - 14 q^{79} - 8 q^{80} - q^{81} - 12 q^{83} - 2 q^{84} - 4 q^{85} - 8 q^{87} - 8 q^{89} - 5 q^{91} - 16 q^{92} - 3 q^{93} + 8 q^{95} + 7 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^4 + 4 * q^5 - q^7 - q^9 + q^11 + 4 * q^12 + 7 * q^13 + 2 * q^15 - 4 * q^16 - 2 * q^17 + 4 * q^19 + 4 * q^20 - 2 * q^21 - 4 * q^23 - 2 * q^25 - 2 * q^27 + 2 * q^28 + 8 * q^29 - 6 * q^31 - q^33 - 2 * q^35 + 2 * q^36 - 10 * q^37 + 2 * q^39 + 8 * q^41 + 7 * q^43 + 4 * q^44 - 2 * q^45 + 16 * q^47 + 4 * q^48 + 6 * q^49 - 4 * q^51 + 10 * q^52 - 20 * q^53 + 2 * q^55 + 8 * q^57 - 8 * q^59 + 8 * q^60 - 13 * q^61 - q^63 - 16 * q^64 + 14 * q^65 - 11 * q^67 + 4 * q^68 + 4 * q^69 + 6 * q^71 - 26 * q^73 - q^75 - 8 * q^76 - 2 * q^77 - 14 * q^79 - 8 * q^80 - q^81 - 12 * q^83 - 2 * q^84 - 4 * q^85 - 8 * q^87 - 8 * q^89 - 5 * q^91 - 16 * q^92 - 3 * q^93 + 8 * q^95 + 7 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$.

$n$	$67$	$79$	$287$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$	2.00000			0.894427			0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000			0.894427			0.447214	+	0.894427i	$0.352416\pi$
$6$		0			0
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i	0.755929	+	0.654654i	$0.227186\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$12$	2.00000			0.577350
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$14$		0			0
$15$	1.00000	+	1.73205i	0.258199	+	0.447214i
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$		0			0
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	−0.540068	−	0.841621i	$-0.681602\pi$
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	0.998899	+	0.0469020i	$0.0149348\pi$
$20$	2.00000	−	3.46410i	0.447214	−	0.774597i
$21$	−1.00000			−0.218218
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$	1.00000	+	1.73205i	0.188982	+	0.327327i
$29$	4.00000	+	6.92820i	0.742781	+	1.28654i	0.951224	+	0.308500i	$0.0998271\pi$
$29$	4.00000	+	6.92820i	0.742781	+	1.28654i	−0.208443	+	0.978035i	$0.566840\pi$
$30$		0			0
$31$	−3.00000			−0.538816			−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000			−0.538816			−0.269408	+	0.963026i	$0.586828\pi$
$32$		0			0
$33$	−0.500000	+	0.866025i	−0.0870388	+	0.150756i
$34$		0			0
$35$	−1.00000	+	1.73205i	−0.169031	+	0.292770i
$36$	1.00000	+	1.73205i	0.166667	+	0.288675i
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	0.0821995	−	0.996616i	$-0.473806\pi$
$38$		0			0
$39$	1.00000	+	3.46410i	0.160128	+	0.554700i
$40$		0			0
$41$	4.00000	+	6.92820i	0.624695	+	1.08200i	0.988600	+	0.150567i	$0.0481100\pi$
$41$	4.00000	+	6.92820i	0.624695	+	1.08200i	−0.363905	+	0.931436i	$0.618557\pi$
$42$		0			0
$43$	3.50000	−	6.06218i	0.533745	−	0.924473i	−0.465478	−	0.885059i	$-0.654118\pi$
$43$	3.50000	−	6.06218i	0.533745	−	0.924473i	0.999223	−	0.0394140i	$-0.0125491\pi$
$44$	2.00000			0.301511
$45$	−1.00000	+	1.73205i	−0.149071	+	0.258199i
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$	2.00000	−	3.46410i	0.288675	−	0.500000i
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$		0			0
$51$	−2.00000			−0.280056
$52$	5.00000	−	5.19615i	0.693375	−	0.720577i
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	1.00000	+	1.73205i	0.134840	+	0.233550i
$56$		0			0
$57$	4.00000			0.529813
$58$		0			0
$59$	−4.00000	+	6.92820i	−0.520756	+	0.901975i	0.478953	+	0.877841i	$0.341016\pi$
$59$	−4.00000	+	6.92820i	−0.520756	+	0.901975i	−0.999709	+	0.0241347i	$0.992317\pi$
$60$	4.00000			0.516398
$61$	−6.50000	+	11.2583i	−0.832240	+	1.44148i	0.0640184	+	0.997949i	$0.479608\pi$
$61$	−6.50000	+	11.2583i	−0.832240	+	1.44148i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	−0.500000	−	0.866025i	−0.0629941	−	0.109109i
$64$	−8.00000			−1.00000
$65$	7.00000	+	1.73205i	0.868243	+	0.214834i
$66$		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	$-0.401202\pi$
$68$	2.00000	+	3.46410i	0.242536	+	0.420084i
$69$	2.00000	−	3.46410i	0.240772	−	0.417029i
$70$		0			0
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	−0.631260	−	0.775571i	$-0.717462\pi$
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	0.987294	+	0.158901i	$0.0507952\pi$
$72$		0			0
$73$	−13.0000			−1.52153			−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000			−1.52153			−0.760767	+	0.649025i	$0.775177\pi$
$74$		0			0
$75$	−0.500000	−	0.866025i	−0.0577350	−	0.100000i
$76$	−4.00000	−	6.92820i	−0.458831	−	0.794719i
$77$	−1.00000			−0.113961
$78$		0			0
$79$	−7.00000			−0.787562			−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000			−0.787562			−0.393781	+	0.919204i	$0.628833\pi$
$80$	−4.00000	−	6.92820i	−0.447214	−	0.774597i
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$	−1.00000	+	1.73205i	−0.109109	+	0.188982i
$85$	−2.00000	+	3.46410i	−0.216930	+	0.375735i
$86$		0			0
$87$	−4.00000	+	6.92820i	−0.428845	+	0.742781i
$88$		0			0
$89$	−4.00000	−	6.92820i	−0.423999	−	0.734388i	0.572327	−	0.820025i	$-0.306041\pi$
$89$	−4.00000	−	6.92820i	−0.423999	−	0.734388i	−0.996326	+	0.0856373i	$0.972707\pi$
$90$		0			0
$91$	−2.50000	+	2.59808i	−0.262071	+	0.272352i
$92$	−8.00000			−0.834058
$93$	−1.50000	−	2.59808i	−0.155543	−	0.269408i
$94$		0			0
$95$	4.00000	−	6.92820i	0.410391	−	0.710819i
$96$		0			0
$97$	3.50000	−	6.06218i	0.355371	−	0.615521i	−0.631810	−	0.775123i	$-0.717688\pi$
$97$	3.50000	−	6.06218i	0.355371	−	0.615521i	0.987181	+	0.159602i	$0.0510211\pi$
$98$		0			0
$99$	−1.00000			−0.100504
$100$	−1.00000	+	1.73205i	−0.100000	+	0.173205i
$101$	3.00000	+	5.19615i	0.298511	+	0.517036i	0.975796	−	0.218685i	$-0.0701767\pi$
$101$	3.00000	+	5.19615i	0.298511	+	0.517036i	−0.677284	+	0.735721i	$0.736843\pi$
$102$		0			0
$103$	11.0000			1.08386			0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000			1.08386			0.541931	+	0.840423i	$0.317693\pi$
$104$		0			0
$105$	−2.00000			−0.195180
$106$		0			0
$107$	−4.00000	−	6.92820i	−0.386695	−	0.669775i	0.605308	−	0.795991i	$-0.293050\pi$
$107$	−4.00000	−	6.92820i	−0.386695	−	0.669775i	−0.992003	+	0.126217i	$0.959717\pi$
$108$	−1.00000	+	1.73205i	−0.0962250	+	0.166667i
$109$	−5.00000			−0.478913			−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000			−0.478913			−0.239457	+	0.970907i	$0.576969\pi$
$110$		0			0
$111$	5.00000	−	8.66025i	0.474579	−	0.821995i
$112$	4.00000			0.377964
$113$	−8.00000	+	13.8564i	−0.752577	+	1.30350i	0.193993	+	0.981003i	$0.437856\pi$
$113$	−8.00000	+	13.8564i	−0.752577	+	1.30350i	−0.946570	+	0.322498i	$0.895477\pi$
$114$		0			0
$115$	−4.00000	−	6.92820i	−0.373002	−	0.646058i
$116$	16.0000			1.48556
$117$	−2.50000	+	2.59808i	−0.231125	+	0.240192i
$118$		0			0
$119$	−1.00000	−	1.73205i	−0.0916698	−	0.158777i
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.0454545	+	0.0787296i
$122$		0			0
$123$	−4.00000	+	6.92820i	−0.360668	+	0.624695i
$124$	−3.00000	+	5.19615i	−0.269408	+	0.466628i
$125$	−12.0000			−1.07331
$126$		0			0
$127$	−0.500000	−	0.866025i	−0.0443678	−	0.0768473i	0.842989	−	0.537931i	$-0.180794\pi$
$127$	−0.500000	−	0.866025i	−0.0443678	−	0.0768473i	−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$	7.00000			0.616316
$130$		0			0
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$	1.00000	+	1.73205i	0.0870388	+	0.150756i
$133$	2.00000	+	3.46410i	0.173422	+	0.300376i
$134$		0			0
$135$	−2.00000			−0.172133
$136$		0			0
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	−0.820141	−	0.572161i	$-0.806105\pi$
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	0.905577	+	0.424182i	$0.139438\pi$
$138$		0			0
$139$	4.50000	−	7.79423i	0.381685	−	0.661098i	−0.609618	−	0.792695i	$-0.708677\pi$
$139$	4.50000	−	7.79423i	0.381685	−	0.661098i	0.991303	+	0.131597i	$0.0420106\pi$
$140$	2.00000	+	3.46410i	0.169031	+	0.292770i
$141$	4.00000	+	6.92820i	0.336861	+	0.583460i
$142$		0			0
$143$	1.00000	+	3.46410i	0.0836242	+	0.289683i
$144$	4.00000			0.333333
$145$	8.00000	+	13.8564i	0.664364	+	1.15071i
$146$		0			0
$147$	−3.00000	+	5.19615i	−0.247436	+	0.428571i
$148$	−20.0000			−1.64399
$149$	5.00000	−	8.66025i	0.409616	−	0.709476i	−0.585231	−	0.810867i	$-0.698996\pi$
$149$	5.00000	−	8.66025i	0.409616	−	0.709476i	0.994847	+	0.101391i	$0.0323294\pi$
$150$		0			0
$151$	4.00000			0.325515			0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000			0.325515			0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$	−1.00000	−	1.73205i	−0.0808452	−	0.140028i
$154$		0			0
$155$	−6.00000			−0.481932
$156$	7.00000	+	1.73205i	0.560449	+	0.138675i
$157$	23.0000			1.83560			0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000			1.83560			0.917800	+	0.397043i	$0.129964\pi$
$158$		0			0
$159$	−5.00000	−	8.66025i	−0.396526	−	0.686803i
$160$		0			0
$161$	4.00000			0.315244
$162$		0			0
$163$	0.500000	−	0.866025i	0.0391630	−	0.0678323i	−0.845780	−	0.533533i	$-0.820864\pi$
$163$	0.500000	−	0.866025i	0.0391630	−	0.0678323i	0.884943	+	0.465700i	$0.154198\pi$
$164$	16.0000			1.24939
$165$	−1.00000	+	1.73205i	−0.0778499	+	0.134840i
$166$		0			0
$167$	7.00000	+	12.1244i	0.541676	+	0.938211i	0.998808	+	0.0488118i	$0.0155435\pi$
$167$	7.00000	+	12.1244i	0.541676	+	0.938211i	−0.457132	+	0.889399i	$0.651123\pi$
$168$		0			0
$169$	11.5000	+	6.06218i	0.884615	+	0.466321i
$170$		0			0
$171$	2.00000	+	3.46410i	0.152944	+	0.264906i
$172$	−7.00000	−	12.1244i	−0.533745	−	0.924473i
$173$	−5.00000	+	8.66025i	−0.380143	+	0.658427i	−0.991082	−	0.133250i	$-0.957459\pi$
$173$	−5.00000	+	8.66025i	−0.380143	+	0.658427i	0.610939	+	0.791677i	$0.290792\pi$
$174$		0			0
$175$	0.500000	−	0.866025i	0.0377964	−	0.0654654i
$176$	2.00000	−	3.46410i	0.150756	−	0.261116i
$177$	−8.00000			−0.601317
$178$		0			0
$179$	12.0000	+	20.7846i	0.896922	+	1.55351i	0.831408	+	0.555663i	$0.187536\pi$
$179$	12.0000	+	20.7846i	0.896922	+	1.55351i	0.0655145	+	0.997852i	$0.479131\pi$
$180$	2.00000	+	3.46410i	0.149071	+	0.258199i
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	−13.0000			−0.960988
$184$		0			0
$185$	−10.0000	−	17.3205i	−0.735215	−	1.27343i
$186$		0			0
$187$	−2.00000			−0.146254
$188$	8.00000	−	13.8564i	0.583460	−	1.01058i
$189$	0.500000	−	0.866025i	0.0363696	−	0.0629941i
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$	−4.00000	−	6.92820i	−0.288675	−	0.500000i
$193$	1.50000	+	2.59808i	0.107972	+	0.187014i	0.914949	−	0.403570i	$-0.132231\pi$
$193$	1.50000	+	2.59808i	0.107972	+	0.187014i	−0.806976	+	0.590584i	$0.798898\pi$
$194$		0			0
$195$	2.00000	+	6.92820i	0.143223	+	0.496139i
$196$	12.0000			0.857143
$197$	−10.0000	−	17.3205i	−0.712470	−	1.23404i	−0.963927	−	0.266167i	$-0.914243\pi$
$197$	−10.0000	−	17.3205i	−0.712470	−	1.23404i	0.251457	−	0.967869i	$-0.419090\pi$
$198$		0			0
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	−0.714893	−	0.699234i	$-0.753524\pi$
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	0.963001	+	0.269498i	$0.0868577\pi$
$200$		0			0
$201$	5.50000	−	9.52628i	0.387940	−	0.671932i
$202$		0			0
$203$	−8.00000			−0.561490
$204$	−2.00000	+	3.46410i	−0.140028	+	0.242536i
$205$	8.00000	+	13.8564i	0.558744	+	0.967773i
$206$		0			0
$207$	4.00000			0.278019
$208$	−4.00000	−	13.8564i	−0.277350	−	0.960769i
$209$	4.00000			0.276686
$210$		0			0
$211$	7.50000	+	12.9904i	0.516321	+	0.894295i	0.999820	+	0.0189499i	$0.00603229\pi$
$211$	7.50000	+	12.9904i	0.516321	+	0.894295i	−0.483499	+	0.875345i	$0.660634\pi$
$212$	−10.0000	+	17.3205i	−0.686803	+	1.18958i
$213$	6.00000			0.411113
$214$		0			0
$215$	7.00000	−	12.1244i	0.477396	−	0.826874i
$216$		0			0
$217$	1.50000	−	2.59808i	0.101827	−	0.176369i
$218$		0			0
$219$	−6.50000	−	11.2583i	−0.439229	−	0.760767i
$220$	4.00000			0.269680
$221$	−5.00000	+	5.19615i	−0.336336	+	0.349531i
$222$		0			0
$223$	2.00000	+	3.46410i	0.133930	+	0.231973i	0.925188	−	0.379509i	$-0.123907\pi$
$223$	2.00000	+	3.46410i	0.133930	+	0.231973i	−0.791258	+	0.611482i	$0.790574\pi$
$224$		0			0
$225$	0.500000	−	0.866025i	0.0333333	−	0.0577350i
$226$		0			0
$227$	12.0000	−	20.7846i	0.796468	−	1.37952i	−0.125435	−	0.992102i	$-0.540033\pi$
$227$	12.0000	−	20.7846i	0.796468	−	1.37952i	0.921903	−	0.387421i	$-0.126634\pi$
$228$	4.00000	−	6.92820i	0.264906	−	0.458831i
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$	−0.500000	−	0.866025i	−0.0328976	−	0.0569803i
$232$		0			0
$233$	8.00000			0.524097			0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000			0.524097			0.262049	+	0.965055i	$0.415602\pi$
$234$		0			0
$235$	16.0000			1.04372
$236$	8.00000	+	13.8564i	0.520756	+	0.901975i
$237$	−3.50000	−	6.06218i	−0.227349	−	0.393781i
$238$		0			0
$239$	−18.0000			−1.16432			−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000			−1.16432			−0.582162	+	0.813073i	$0.697793\pi$
$240$	4.00000	−	6.92820i	0.258199	−	0.447214i
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	0.980917	+	0.194429i	$0.0622852\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.0320750	−	0.0555556i
$244$	13.0000	+	22.5167i	0.832240	+	1.44148i
$245$	6.00000	+	10.3923i	0.383326	+	0.663940i
$246$		0			0
$247$	10.0000	−	10.3923i	0.636285	−	0.661247i
$248$		0			0
$249$	−3.00000	−	5.19615i	−0.190117	−	0.329293i
$250$		0			0
$251$	9.00000	−	15.5885i	0.568075	−	0.983935i	−0.428681	−	0.903456i	$-0.641022\pi$
$251$	9.00000	−	15.5885i	0.568075	−	0.983935i	0.996756	−	0.0804789i	$-0.0256450\pi$
$252$	−2.00000			−0.125988
$253$	2.00000	−	3.46410i	0.125739	−	0.217786i
$254$		0			0
$255$	−4.00000			−0.250490
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	0.990217	−	0.139533i	$-0.0445601\pi$
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	−0.615948	+	0.787787i	$0.711227\pi$
$258$		0			0
$259$	10.0000			0.621370
$260$	10.0000	−	10.3923i	0.620174	−	0.644503i
$261$	−8.00000			−0.495188
$262$		0			0
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	0.943572	−	0.331166i	$-0.107442\pi$
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	−0.758585	+	0.651575i	$0.774109\pi$
$264$		0			0
$265$	−20.0000			−1.22859
$266$		0			0
$267$	4.00000	−	6.92820i	0.244796	−	0.423999i
$268$	−22.0000			−1.34386
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	8.50000	+	14.7224i	0.516338	+	0.894324i	0.999820	+	0.0189696i	$0.00603859\pi$
$271$	8.50000	+	14.7224i	0.516338	+	0.894324i	−0.483482	+	0.875354i	$0.660628\pi$
$272$	8.00000			0.485071
$273$	−3.50000	−	0.866025i	−0.211830	−	0.0524142i
$274$		0			0
$275$	−0.500000	−	0.866025i	−0.0301511	−	0.0522233i
$276$	−4.00000	−	6.92820i	−0.240772	−	0.417029i
$277$	−1.00000	+	1.73205i	−0.0600842	+	0.104069i	−0.894503	−	0.447062i	$-0.852470\pi$
$277$	−1.00000	+	1.73205i	−0.0600842	+	0.104069i	0.834419	+	0.551131i	$0.185804\pi$
$278$		0			0
$279$	1.50000	−	2.59808i	0.0898027	−	0.155543i
$280$		0			0
$281$	24.0000			1.43172			0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000			1.43172			0.715860	+	0.698244i	$0.246035\pi$
$282$		0			0
$283$	1.50000	+	2.59808i	0.0891657	+	0.154440i	0.907159	−	0.420789i	$-0.138247\pi$
$283$	1.50000	+	2.59808i	0.0891657	+	0.154440i	−0.817993	+	0.575228i	$0.804913\pi$
$284$	−6.00000	−	10.3923i	−0.356034	−	0.616670i
$285$	8.00000			0.473879
$286$		0			0
$287$	−8.00000			−0.472225
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$	7.00000			0.410347
$292$	−13.0000	+	22.5167i	−0.760767	+	1.31769i
$293$	8.00000	−	13.8564i	0.467365	−	0.809500i	−0.531940	−	0.846782i	$-0.678537\pi$
$293$	8.00000	−	13.8564i	0.467365	−	0.809500i	0.999305	+	0.0372823i	$0.0118701\pi$
$294$		0			0
$295$	−8.00000	+	13.8564i	−0.465778	+	0.806751i
$296$		0			0
$297$	−0.500000	−	0.866025i	−0.0290129	−	0.0502519i
$298$		0			0
$299$	−4.00000	−	13.8564i	−0.231326	−	0.801337i
$300$	−2.00000			−0.115470
$301$	3.50000	+	6.06218i	0.201737	+	0.349418i
$302$		0			0
$303$	−3.00000	+	5.19615i	−0.172345	+	0.298511i
$304$	−16.0000			−0.917663
$305$	−13.0000	+	22.5167i	−0.744378	+	1.28930i
$306$		0			0
$307$	9.00000			0.513657			0.256829	−	0.966457i	$-0.417322\pi$
$307$	9.00000			0.513657			0.256829	+	0.966457i	$0.417322\pi$
$308$	−1.00000	+	1.73205i	−0.0569803	+	0.0986928i
$309$	5.50000	+	9.52628i	0.312884	+	0.541931i
$310$		0			0
$311$	−16.0000			−0.907277			−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000			−0.907277			−0.453638	+	0.891186i	$0.649874\pi$
$312$		0			0
$313$	−9.00000			−0.508710			−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000			−0.508710			−0.254355	+	0.967111i	$0.581863\pi$
$314$		0			0
$315$	−1.00000	−	1.73205i	−0.0563436	−	0.0975900i
$316$	−7.00000	+	12.1244i	−0.393781	+	0.682048i
$317$	14.0000			0.786318			0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000			0.786318			0.393159	+	0.919470i	$0.371382\pi$
$318$		0			0
$319$	−4.00000	+	6.92820i	−0.223957	+	0.387905i
$320$	−16.0000			−0.894427
$321$	4.00000	−	6.92820i	0.223258	−	0.386695i
$322$		0			0
$323$	4.00000	+	6.92820i	0.222566	+	0.385496i
$324$	−2.00000			−0.111111
$325$	−3.50000	−	0.866025i	−0.194145	−	0.0480384i
$326$		0			0
$327$	−2.50000	−	4.33013i	−0.138250	−	0.239457i
$328$		0			0
$329$	−4.00000	+	6.92820i	−0.220527	+	0.381964i
$330$		0			0
$331$	11.5000	−	19.9186i	0.632097	−	1.09482i	−0.355025	−	0.934857i	$-0.615528\pi$
$331$	11.5000	−	19.9186i	0.632097	−	1.09482i	0.987122	−	0.159968i	$-0.0511390\pi$
$332$	−6.00000	+	10.3923i	−0.329293	+	0.570352i
$333$	10.0000			0.547997
$334$		0			0
$335$	−11.0000	−	19.0526i	−0.600994	−	1.04095i
$336$	2.00000	+	3.46410i	0.109109	+	0.188982i
$337$	7.00000			0.381314			0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000			0.381314			0.190657	+	0.981657i	$0.438938\pi$
$338$		0			0
$339$	−16.0000			−0.869001
$340$	4.00000	+	6.92820i	0.216930	+	0.375735i
$341$	−1.50000	−	2.59808i	−0.0812296	−	0.140694i
$342$		0			0
$343$	−13.0000			−0.701934
$344$		0			0
$345$	4.00000	−	6.92820i	0.215353	−	0.373002i
$346$		0			0
$347$	9.00000	−	15.5885i	0.483145	−	0.836832i	−0.516667	−	0.856186i	$-0.672828\pi$
$347$	9.00000	−	15.5885i	0.483145	−	0.836832i	0.999813	+	0.0193540i	$0.00616095\pi$
$348$	8.00000	+	13.8564i	0.428845	+	0.742781i
$349$	5.50000	+	9.52628i	0.294408	+	0.509930i	0.974847	−	0.222875i	$-0.0715441\pi$
$349$	5.50000	+	9.52628i	0.294408	+	0.509930i	−0.680439	+	0.732805i	$0.738211\pi$
$350$		0			0
$351$	−3.50000	−	0.866025i	−0.186816	−	0.0462250i
$352$		0			0
$353$	−9.00000	−	15.5885i	−0.479022	−	0.829690i	0.520689	−	0.853746i	$-0.325675\pi$
$353$	−9.00000	−	15.5885i	−0.479022	−	0.829690i	−0.999711	+	0.0240566i	$0.992342\pi$
$354$		0			0
$355$	6.00000	−	10.3923i	0.318447	−	0.551566i
$356$	−16.0000			−0.847998
$357$	1.00000	−	1.73205i	0.0529256	−	0.0916698i
$358$		0			0
$359$	−20.0000			−1.05556			−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000			−1.05556			−0.527780	+	0.849381i	$0.676975\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$	−1.00000			−0.0524864
$364$	2.00000	+	6.92820i	0.104828	+	0.363137i
$365$	−26.0000			−1.36090
$366$		0			0
$367$	−4.50000	−	7.79423i	−0.234898	−	0.406855i	0.724345	−	0.689438i	$-0.242142\pi$
$367$	−4.50000	−	7.79423i	−0.234898	−	0.406855i	−0.959243	+	0.282582i	$0.908809\pi$
$368$	−8.00000	+	13.8564i	−0.417029	+	0.722315i
$369$	−8.00000			−0.416463
$370$		0			0
$371$	5.00000	−	8.66025i	0.259587	−	0.449618i
$372$	−6.00000			−0.311086
$373$	−14.5000	+	25.1147i	−0.750782	+	1.30039i	0.196663	+	0.980471i	$0.436990\pi$
$373$	−14.5000	+	25.1147i	−0.750782	+	1.30039i	−0.947444	+	0.319921i	$0.896344\pi$
$374$		0			0
$375$	−6.00000	−	10.3923i	−0.309839	−	0.536656i
$376$		0			0
$377$	8.00000	+	27.7128i	0.412021	+	1.42728i
$378$		0			0
$379$	14.5000	+	25.1147i	0.744815	+	1.29006i	0.950281	+	0.311393i	$0.100796\pi$
$379$	14.5000	+	25.1147i	0.744815	+	1.29006i	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−8.00000	−	13.8564i	−0.410391	−	0.710819i
$381$	0.500000	−	0.866025i	0.0256158	−	0.0443678i
$382$		0			0
$383$	8.00000	−	13.8564i	0.408781	−	0.708029i	−0.585973	−	0.810331i	$-0.699287\pi$
$383$	8.00000	−	13.8564i	0.408781	−	0.708029i	0.994753	+	0.102302i	$0.0326207\pi$
$384$		0			0
$385$	−2.00000			−0.101929
$386$		0			0
$387$	3.50000	+	6.06218i	0.177915	+	0.308158i
$388$	−7.00000	−	12.1244i	−0.355371	−	0.615521i
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$	−9.00000	−	15.5885i	−0.453990	−	0.786334i
$394$		0			0
$395$	−14.0000			−0.704416
$396$	−1.00000	+	1.73205i	−0.0502519	+	0.0870388i
$397$	−1.50000	+	2.59808i	−0.0752828	+	0.130394i	−0.901209	−	0.433384i	$-0.857319\pi$
$397$	−1.50000	+	2.59808i	−0.0752828	+	0.130394i	0.825926	+	0.563778i	$0.190653\pi$
$398$		0			0
$399$	−2.00000	+	3.46410i	−0.100125	+	0.173422i
$400$	2.00000	+	3.46410i	0.100000	+	0.173205i
$401$	−3.00000	−	5.19615i	−0.149813	−	0.259483i	0.781345	−	0.624099i	$-0.214534\pi$
$401$	−3.00000	−	5.19615i	−0.149813	−	0.259483i	−0.931158	+	0.364615i	$0.881200\pi$
$402$		0			0
$403$	−10.5000	−	2.59808i	−0.523042	−	0.129419i
$404$	12.0000			0.597022
$405$	−1.00000	−	1.73205i	−0.0496904	−	0.0860663i
$406$		0			0
$407$	5.00000	−	8.66025i	0.247841	−	0.429273i
$408$		0			0
$409$	16.5000	−	28.5788i	0.815872	−	1.41313i	−0.0928272	−	0.995682i	$-0.529590\pi$
$409$	16.5000	−	28.5788i	0.815872	−	1.41313i	0.908700	−	0.417450i	$-0.137076\pi$
$410$		0			0
$411$	2.00000			0.0986527
$412$	11.0000	−	19.0526i	0.541931	−	0.938652i
$413$	−4.00000	−	6.92820i	−0.196827	−	0.340915i
$414$		0			0
$415$	−12.0000			−0.589057
$416$		0			0
$417$	9.00000			0.440732
$418$		0			0
$419$	−13.0000	−	22.5167i	−0.635092	−	1.10001i	−0.986496	−	0.163787i	$-0.947629\pi$
$419$	−13.0000	−	22.5167i	−0.635092	−	1.10001i	0.351404	−	0.936224i	$-0.385704\pi$
$420$	−2.00000	+	3.46410i	−0.0975900	+	0.169031i
$421$	5.00000			0.243685			0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000			0.243685			0.121843	+	0.992549i	$0.461120\pi$
$422$		0			0
$423$	−4.00000	+	6.92820i	−0.194487	+	0.336861i
$424$		0			0
$425$	1.00000	−	1.73205i	0.0485071	−	0.0840168i
$426$		0			0
$427$	−6.50000	−	11.2583i	−0.314557	−	0.544829i
$428$	−16.0000			−0.773389
$429$	−2.50000	+	2.59808i	−0.120701	+	0.125436i
$430$		0			0
$431$	11.0000	+	19.0526i	0.529851	+	0.917729i	0.999394	+	0.0348195i	$0.0110856\pi$
$431$	11.0000	+	19.0526i	0.529851	+	0.917729i	−0.469542	+	0.882910i	$0.655581\pi$
$432$	2.00000	+	3.46410i	0.0962250	+	0.166667i
$433$	10.5000	−	18.1865i	0.504598	−	0.873989i	−0.495388	−	0.868672i	$-0.664974\pi$
$433$	10.5000	−	18.1865i	0.504598	−	0.873989i	0.999986	−	0.00531724i	$-0.00169254\pi$
$434$		0			0
$435$	−8.00000	+	13.8564i	−0.383571	+	0.664364i
$436$	−5.00000	+	8.66025i	−0.239457	+	0.414751i
$437$	−16.0000			−0.765384
$438$		0			0
$439$	−10.5000	−	18.1865i	−0.501138	−	0.867996i	−0.999999	−	0.00131415i	$-0.999582\pi$
$439$	−10.5000	−	18.1865i	−0.501138	−	0.867996i	0.498861	−	0.866682i	$-0.333752\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$		0			0
$443$	−20.0000			−0.950229			−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000			−0.950229			−0.475114	+	0.879924i	$0.657593\pi$
$444$	−10.0000	−	17.3205i	−0.474579	−	0.821995i
$445$	−8.00000	−	13.8564i	−0.379236	−	0.656857i
$446$		0			0
$447$	10.0000			0.472984
$448$	4.00000	−	6.92820i	0.188982	−	0.327327i
$449$	6.00000	−	10.3923i	0.283158	−	0.490443i	−0.689003	−	0.724758i	$-0.741951\pi$
$449$	6.00000	−	10.3923i	0.283158	−	0.490443i	0.972161	+	0.234315i	$0.0752847\pi$
$450$		0			0
$451$	−4.00000	+	6.92820i	−0.188353	+	0.326236i
$452$	16.0000	+	27.7128i	0.752577	+	1.30350i
$453$	2.00000	+	3.46410i	0.0939682	+	0.162758i
$454$		0			0
$455$	−5.00000	+	5.19615i	−0.234404	+	0.243599i
$456$		0			0
$457$	18.5000	+	32.0429i	0.865393	+	1.49891i	0.866656	+	0.498906i	$0.166265\pi$
$457$	18.5000	+	32.0429i	0.865393	+	1.49891i	−0.00126243	+	0.999999i	$0.500402\pi$
$458$		0			0
$459$	1.00000	−	1.73205i	0.0466760	−	0.0808452i
$460$	−16.0000			−0.746004
$461$	1.00000	−	1.73205i	0.0465746	−	0.0806696i	−0.841798	−	0.539792i	$-0.818503\pi$
$461$	1.00000	−	1.73205i	0.0465746	−	0.0806696i	0.888373	+	0.459123i	$0.151836\pi$
$462$		0			0
$463$	−27.0000			−1.25480			−0.627398	−	0.778699i	$-0.715880\pi$
$463$	−27.0000			−1.25480			−0.627398	+	0.778699i	$0.715880\pi$
$464$	16.0000	−	27.7128i	0.742781	−	1.28654i
$465$	−3.00000	−	5.19615i	−0.139122	−	0.240966i
$466$		0			0
$467$	−2.00000			−0.0925490			−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000			−0.0925490			−0.0462745	+	0.998929i	$0.514735\pi$
$468$	2.00000	+	6.92820i	0.0924500	+	0.320256i
$469$	11.0000			0.507933
$470$		0			0
$471$	11.5000	+	19.9186i	0.529892	+	0.917800i
$472$		0			0
$473$	7.00000			0.321860
$474$		0			0
$475$	−2.00000	+	3.46410i	−0.0917663	+	0.158944i
$476$	−4.00000			−0.183340
$477$	5.00000	−	8.66025i	0.228934	−	0.396526i
$478$		0			0
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	0.695773	−	0.718262i	$-0.255062\pi$
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	−0.969920	+	0.243426i	$0.921729\pi$
$480$		0			0
$481$	−10.0000	−	34.6410i	−0.455961	−	1.57949i
$482$		0			0
$483$	2.00000	+	3.46410i	0.0910032	+	0.157622i
$484$	1.00000	+	1.73205i	0.0454545	+	0.0787296i
$485$	7.00000	−	12.1244i	0.317854	−	0.550539i
$486$		0			0
$487$	16.0000	−	27.7128i	0.725029	−	1.25579i	−0.233933	−	0.972253i	$-0.575160\pi$
$487$	16.0000	−	27.7128i	0.725029	−	1.25579i	0.958962	−	0.283535i	$-0.0915071\pi$
$488$		0			0
$489$	1.00000			0.0452216
$490$		0			0
$491$	15.0000	+	25.9808i	0.676941	+	1.17250i	0.975898	+	0.218229i	$0.0700279\pi$
$491$	15.0000	+	25.9808i	0.676941	+	1.17250i	−0.298957	+	0.954267i	$0.596639\pi$
$492$	8.00000	+	13.8564i	0.360668	+	0.624695i
$493$	−16.0000			−0.720604
$494$		0			0
$495$	−2.00000			−0.0898933
$496$	6.00000	+	10.3923i	0.269408	+	0.466628i
$497$	3.00000	+	5.19615i	0.134568	+	0.233079i
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$	−12.0000	+	20.7846i	−0.536656	+	0.929516i
$501$	−7.00000	+	12.1244i	−0.312737	+	0.541676i
$502$		0			0
$503$	−18.0000	+	31.1769i	−0.802580	+	1.39011i	0.115332	+	0.993327i	$0.463207\pi$
$503$	−18.0000	+	31.1769i	−0.802580	+	1.39011i	−0.917912	+	0.396783i	$0.870127\pi$
$504$		0			0
$505$	6.00000	+	10.3923i	0.266996	+	0.462451i
$506$		0			0
$507$	0.500000	+	12.9904i	0.0222058	+	0.576923i
$508$	−2.00000			−0.0887357
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	6.50000	−	11.2583i	0.287543	−	0.498039i
$512$		0			0
$513$	−2.00000	+	3.46410i	−0.0883022	+	0.152944i
$514$		0			0
$515$	22.0000			0.969436
$516$	7.00000	−	12.1244i	0.308158	−	0.533745i
$517$	4.00000	+	6.92820i	0.175920	+	0.304702i
$518$		0			0
$519$	−10.0000			−0.438951
$520$		0			0
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$	−8.00000	−	13.8564i	−0.349816	−	0.605898i	0.636401	−	0.771358i	$-0.280422\pi$
$523$	−8.00000	−	13.8564i	−0.349816	−	0.605898i	−0.986216	+	0.165460i	$0.947089\pi$
$524$	−18.0000	+	31.1769i	−0.786334	+	1.36197i
$525$	1.00000			0.0436436
$526$		0			0
$527$	3.00000	−	5.19615i	0.130682	−	0.226348i
$528$	4.00000			0.174078
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	−4.00000	−	6.92820i	−0.173585	−	0.300658i
$532$	8.00000			0.346844
$533$	8.00000	+	27.7128i	0.346518	+	1.20038i
$534$		0			0
$535$	−8.00000	−	13.8564i	−0.345870	−	0.599065i
$536$		0			0
$537$	−12.0000	+	20.7846i	−0.517838	+	0.896922i
$538$		0			0
$539$	−3.00000	+	5.19615i	−0.129219	+	0.223814i
$540$	−2.00000	+	3.46410i	−0.0860663	+	0.149071i
$541$	−21.0000			−0.902861			−0.451430	−	0.892306i	$-0.649086\pi$
$541$	−21.0000			−0.902861			−0.451430	+	0.892306i	$0.649086\pi$
$542$		0			0
$543$	−1.00000	−	1.73205i	−0.0429141	−	0.0743294i
$544$		0			0
$545$	−10.0000			−0.428353
$546$		0			0
$547$	−1.00000			−0.0427569			−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000			−0.0427569			−0.0213785	+	0.999771i	$0.506805\pi$
$548$	−2.00000	−	3.46410i	−0.0854358	−	0.147979i
$549$	−6.50000	−	11.2583i	−0.277413	−	0.480494i
$550$		0			0
$551$	32.0000			1.36325
$552$		0			0
$553$	3.50000	−	6.06218i	0.148835	−	0.257790i
$554$		0			0
$555$	10.0000	−	17.3205i	0.424476	−	0.735215i
$556$	−9.00000	−	15.5885i	−0.381685	−	0.661098i
$557$	3.00000	+	5.19615i	0.127114	+	0.220168i	0.922557	−	0.385860i	$-0.126095\pi$
$557$	3.00000	+	5.19615i	0.127114	+	0.220168i	−0.795443	+	0.606028i	$0.792762\pi$
$558$		0			0
$559$	17.5000	−	18.1865i	0.740171	−	0.769208i
$560$	8.00000			0.338062
$561$	−1.00000	−	1.73205i	−0.0422200	−	0.0731272i
$562$		0			0
$563$	−9.00000	+	15.5885i	−0.379305	+	0.656975i	−0.990961	−	0.134148i	$-0.957170\pi$
$563$	−9.00000	+	15.5885i	−0.379305	+	0.656975i	0.611656	+	0.791123i	$0.290503\pi$
$564$	16.0000			0.673722
$565$	−16.0000	+	27.7128i	−0.673125	+	1.16589i
$566$		0			0
$567$	1.00000			0.0419961
$568$		0			0
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	0.741981	−	0.670421i	$-0.233886\pi$
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	−0.951592	+	0.307364i	$0.900553\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$	7.00000	+	1.73205i	0.292685	+	0.0724207i
$573$		0			0
$574$		0			0
$575$	2.00000	+	3.46410i	0.0834058	+	0.144463i
$576$	4.00000	−	6.92820i	0.166667	−	0.288675i
$577$	22.0000			0.915872			0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000			0.915872			0.457936	+	0.888985i	$0.348589\pi$
$578$		0			0
$579$	−1.50000	+	2.59808i	−0.0623379	+	0.107972i
$580$	32.0000			1.32873
$581$	3.00000	−	5.19615i	0.124461	−	0.215573i
$582$		0			0
$583$	−5.00000	−	8.66025i	−0.207079	−	0.358671i
$584$		0			0
$585$	−5.00000	+	5.19615i	−0.206725	+	0.214834i
$586$		0			0
$587$	12.0000	+	20.7846i	0.495293	+	0.857873i	0.999985	−	0.00542667i	$-0.00172737\pi$
$587$	12.0000	+	20.7846i	0.495293	+	0.857873i	−0.504692	+	0.863299i	$0.668394\pi$
$588$	6.00000	+	10.3923i	0.247436	+	0.428571i
$589$	−6.00000	+	10.3923i	−0.247226	+	0.428207i
$590$		0			0
$591$	10.0000	−	17.3205i	0.411345	−	0.712470i
$592$	−20.0000	+	34.6410i	−0.821995	+	1.42374i
$593$	36.0000			1.47834			0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000			1.47834			0.739171	+	0.673517i	$0.235217\pi$
$594$		0			0
$595$	−2.00000	−	3.46410i	−0.0819920	−	0.142014i
$596$	−10.0000	−	17.3205i	−0.409616	−	0.709476i
$597$	7.00000			0.286491
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	23.0000	+	39.8372i	0.938190	+	1.62499i	0.768845	+	0.639435i	$0.220832\pi$
$601$	23.0000	+	39.8372i	0.938190	+	1.62499i	0.169344	+	0.985557i	$0.445835\pi$
$602$		0			0
$603$	11.0000			0.447955
$604$	4.00000	−	6.92820i	0.162758	−	0.281905i
$605$	−1.00000	+	1.73205i	−0.0406558	+	0.0704179i
$606$		0			0
$607$	−4.00000	+	6.92820i	−0.162355	+	0.281207i	−0.935713	−	0.352763i	$-0.885242\pi$
$607$	−4.00000	+	6.92820i	−0.162355	+	0.281207i	0.773358	+	0.633970i	$0.218576\pi$
$608$		0			0
$609$	−4.00000	−	6.92820i	−0.162088	−	0.280745i
$610$		0			0
$611$	28.0000	+	6.92820i	1.13276	+	0.280285i
$612$	−4.00000			−0.161690
$613$	−6.50000	−	11.2583i	−0.262533	−	0.454720i	0.704382	−	0.709821i	$-0.251224\pi$
$613$	−6.50000	−	11.2583i	−0.262533	−	0.454720i	−0.966914	+	0.255102i	$0.917891\pi$
$614$		0			0
$615$	−8.00000	+	13.8564i	−0.322591	+	0.558744i
$616$		0			0
$617$	−12.0000	+	20.7846i	−0.483102	+	0.836757i	−0.999812	−	0.0194037i	$-0.993823\pi$
$617$	−12.0000	+	20.7846i	−0.483102	+	0.836757i	0.516710	+	0.856161i	$0.327157\pi$
$618$		0			0
$619$	17.0000			0.683288			0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000			0.683288			0.341644	+	0.939829i	$0.389016\pi$
$620$	−6.00000	+	10.3923i	−0.240966	+	0.417365i
$621$	2.00000	+	3.46410i	0.0802572	+	0.139010i
$622$		0			0
$623$	8.00000			0.320513
$624$	10.0000	−	10.3923i	0.400320	−	0.416025i
$625$	−19.0000			−0.760000
$626$		0			0
$627$	2.00000	+	3.46410i	0.0798723	+	0.138343i
$628$	23.0000	−	39.8372i	0.917800	−	1.58968i
$629$	20.0000			0.797452
$630$		0			0
$631$	16.5000	−	28.5788i	0.656855	−	1.13771i	−0.324571	−	0.945861i	$-0.605220\pi$
$631$	16.5000	−	28.5788i	0.656855	−	1.13771i	0.981425	−	0.191844i	$-0.0614468\pi$
$632$		0			0
$633$	−7.50000	+	12.9904i	−0.298098	+	0.516321i
$634$		0			0
$635$	−1.00000	−	1.73205i	−0.0396838	−	0.0687343i
$636$	−20.0000			−0.793052
$637$	6.00000	+	20.7846i	0.237729	+	0.823516i
$638$		0			0
$639$	3.00000	+	5.19615i	0.118678	+	0.205557i
$640$		0			0
$641$	24.0000	−	41.5692i	0.947943	−	1.64189i	0.198194	−	0.980163i	$-0.436492\pi$
$641$	24.0000	−	41.5692i	0.947943	−	1.64189i	0.749749	−	0.661723i	$-0.230174\pi$
$642$		0			0
$643$	18.5000	−	32.0429i	0.729569	−	1.26365i	−0.227497	−	0.973779i	$-0.573054\pi$
$643$	18.5000	−	32.0429i	0.729569	−	1.26365i	0.957066	−	0.289871i	$-0.0936125\pi$
$644$	4.00000	−	6.92820i	0.157622	−	0.273009i
$645$	14.0000			0.551249
$646$		0			0
$647$	−9.00000	−	15.5885i	−0.353827	−	0.612845i	0.633090	−	0.774078i	$-0.281786\pi$
$647$	−9.00000	−	15.5885i	−0.353827	−	0.612845i	−0.986916	+	0.161233i	$0.948453\pi$
$648$		0			0
$649$	−8.00000			−0.314027
$650$		0			0
$651$	3.00000			0.117579
$652$	−1.00000	−	1.73205i	−0.0391630	−	0.0678323i
$653$	−13.0000	−	22.5167i	−0.508729	−	0.881145i	−0.999949	−	0.0101092i	$-0.996782\pi$
$653$	−13.0000	−	22.5167i	−0.508729	−	0.881145i	0.491220	−	0.871036i	$-0.336551\pi$
$654$		0			0
$655$	−36.0000			−1.40664
$656$	16.0000	−	27.7128i	0.624695	−	1.08200i
$657$	6.50000	−	11.2583i	0.253589	−	0.439229i
$658$		0			0
$659$	−2.00000	+	3.46410i	−0.0779089	+	0.134942i	−0.902348	−	0.431009i	$-0.858158\pi$
$659$	−2.00000	+	3.46410i	−0.0779089	+	0.134942i	0.824439	+	0.565951i	$0.191491\pi$
$660$	2.00000	+	3.46410i	0.0778499	+	0.134840i
$661$	8.50000	+	14.7224i	0.330612	+	0.572636i	0.982632	−	0.185565i	$-0.0594116\pi$
$661$	8.50000	+	14.7224i	0.330612	+	0.572636i	−0.652020	+	0.758202i	$0.726078\pi$
$662$		0			0
$663$	−7.00000	−	1.73205i	−0.271857	−	0.0672673i
$664$		0			0
$665$	4.00000	+	6.92820i	0.155113	+	0.268664i
$666$		0			0
$667$	16.0000	−	27.7128i	0.619522	−	1.07304i
$668$	28.0000			1.08335
$669$	−2.00000	+	3.46410i	−0.0773245	+	0.133930i
$670$		0			0
$671$	−13.0000			−0.501859
$672$		0			0
$673$	−24.5000	−	42.4352i	−0.944406	−	1.63576i	−0.756937	−	0.653488i	$-0.773305\pi$
$673$	−24.5000	−	42.4352i	−0.944406	−	1.63576i	−0.187469	−	0.982271i	$-0.560028\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$	22.0000	−	13.8564i	0.846154	−	0.532939i
$677$	30.0000			1.15299			0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000			1.15299			0.576497	+	0.817099i	$0.304419\pi$
$678$		0			0
$679$	3.50000	+	6.06218i	0.134318	+	0.232645i
$680$		0			0
$681$	24.0000			0.919682
$682$		0			0
$683$	−13.0000	+	22.5167i	−0.497431	+	0.861576i	−0.999996	−	0.00296369i	$-0.999057\pi$
$683$	−13.0000	+	22.5167i	−0.497431	+	0.861576i	0.502564	+	0.864540i	$0.332390\pi$
$684$	8.00000			0.305888
$685$	2.00000	−	3.46410i	0.0764161	−	0.132357i
$686$		0			0
$687$	3.00000	+	5.19615i	0.114457	+	0.198246i
$688$	−28.0000			−1.06749
$689$	−35.0000	−	8.66025i	−1.33339	−	0.329929i
$690$		0			0
$691$	1.50000	+	2.59808i	0.0570627	+	0.0988355i	0.893146	−	0.449768i	$-0.148493\pi$
$691$	1.50000	+	2.59808i	0.0570627	+	0.0988355i	−0.836083	+	0.548603i	$0.815160\pi$
$692$	10.0000	+	17.3205i	0.380143	+	0.658427i
$693$	0.500000	−	0.866025i	0.0189934	−	0.0328976i
$694$		0			0
$695$	9.00000	−	15.5885i	0.341389	−	0.591304i
$696$		0			0
$697$	−16.0000			−0.606043
$698$		0			0
$699$	4.00000	+	6.92820i	0.151294	+	0.262049i
$700$	−1.00000	−	1.73205i	−0.0377964	−	0.0654654i
$701$	38.0000			1.43524			0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000			1.43524			0.717620	+	0.696435i	$0.245231\pi$
$702$		0			0
$703$	−40.0000			−1.50863
$704$	−4.00000	−	6.92820i	−0.150756	−	0.261116i
$705$	8.00000	+	13.8564i	0.301297	+	0.521862i
$706$		0			0
$707$	−6.00000			−0.225653
$708$	−8.00000	+	13.8564i	−0.300658	+	0.520756i
$709$	−6.50000	+	11.2583i	−0.244113	+	0.422815i	−0.961882	−	0.273466i	$-0.911830\pi$
$709$	−6.50000	+	11.2583i	−0.244113	+	0.422815i	0.717769	+	0.696281i	$0.245163\pi$
$710$		0			0
$711$	3.50000	−	6.06218i	0.131260	−	0.227349i
$712$		0			0
$713$	6.00000	+	10.3923i	0.224702	+	0.389195i
$714$		0			0
$715$	2.00000	+	6.92820i	0.0747958	+	0.259100i
$716$	48.0000			1.79384
$717$	−9.00000	−	15.5885i	−0.336111	−	0.582162i
$718$		0			0
$719$	−8.00000	+	13.8564i	−0.298350	+	0.516757i	−0.975759	−	0.218850i	$-0.929769\pi$
$719$	−8.00000	+	13.8564i	−0.298350	+	0.516757i	0.677409	+	0.735607i	$0.263103\pi$
$720$	8.00000			0.298142
$721$	−5.50000	+	9.52628i	−0.204831	+	0.354777i
$722$		0			0
$723$	10.0000			0.371904
$724$	−2.00000	+	3.46410i	−0.0743294	+	0.128742i
$725$	−4.00000	−	6.92820i	−0.148556	−	0.257307i
$726$		0			0
$727$	−5.00000			−0.185440			−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000			−0.185440			−0.0927199	+	0.995692i	$0.529556\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	7.00000	+	12.1244i	0.258904	+	0.448435i
$732$	−13.0000	+	22.5167i	−0.480494	+	0.832240i
$733$	−1.00000			−0.0369358			−0.0184679	−	0.999829i	$-0.505879\pi$
$733$	−1.00000			−0.0369358			−0.0184679	+	0.999829i	$0.505879\pi$
$734$		0			0
$735$	−6.00000	+	10.3923i	−0.221313	+	0.383326i
$736$		0			0
$737$	5.50000	−	9.52628i	0.202595	−	0.350905i
$738$		0			0
$739$	10.0000	+	17.3205i	0.367856	+	0.637145i	0.989230	−	0.146369i	$-0.0467586\pi$
$739$	10.0000	+	17.3205i	0.367856	+	0.637145i	−0.621374	+	0.783514i	$0.713425\pi$
$740$	−40.0000			−1.47043
$741$	14.0000	+	3.46410i	0.514303	+	0.127257i
$742$		0			0
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	0.805735	−	0.592277i	$-0.201771\pi$
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	−0.915794	+	0.401648i	$0.868437\pi$
$744$		0			0
$745$	10.0000	−	17.3205i	0.366372	−	0.634574i
$746$		0			0
$747$	3.00000	−	5.19615i	0.109764	−	0.190117i
$748$	−2.00000	+	3.46410i	−0.0731272	+	0.126660i
$749$	8.00000			0.292314
$750$		0			0
$751$	−24.0000	−	41.5692i	−0.875772	−	1.51688i	−0.855938	−	0.517079i	$-0.827019\pi$
$751$	−24.0000	−	41.5692i	−0.875772	−	1.51688i	−0.0198348	−	0.999803i	$-0.506314\pi$
$752$	−16.0000	−	27.7128i	−0.583460	−	1.01058i
$753$	18.0000			0.655956
$754$		0			0
$755$	8.00000			0.291150
$756$	−1.00000	−	1.73205i	−0.0363696	−	0.0629941i
$757$	27.0000	+	46.7654i	0.981332	+	1.69972i	0.657222	+	0.753697i	$0.271731\pi$
$757$	27.0000	+	46.7654i	0.981332	+	1.69972i	0.324109	+	0.946020i	$0.394935\pi$
$758$		0			0
$759$	4.00000			0.145191
$760$		0			0
$761$	−7.00000	+	12.1244i	−0.253750	+	0.439508i	−0.964555	−	0.263881i	$-0.914997\pi$
$761$	−7.00000	+	12.1244i	−0.253750	+	0.439508i	0.710805	+	0.703389i	$0.248331\pi$
$762$		0			0
$763$	2.50000	−	4.33013i	0.0905061	−	0.156761i
$764$		0			0
$765$	−2.00000	−	3.46410i	−0.0723102	−	0.125245i
$766$		0			0
$767$	−20.0000	+	20.7846i	−0.722158	+	0.750489i
$768$	−16.0000			−0.577350
$769$	15.0000	+	25.9808i	0.540914	+	0.936890i	0.998852	+	0.0479061i	$0.0152548\pi$
$769$	15.0000	+	25.9808i	0.540914	+	0.936890i	−0.457938	+	0.888984i	$0.651412\pi$
$770$		0			0
$771$	−6.00000	+	10.3923i	−0.216085	+	0.374270i
$772$	6.00000			0.215945
$773$	−15.0000	+	25.9808i	−0.539513	+	0.934463i	0.459418	+	0.888220i	$0.348058\pi$
$773$	−15.0000	+	25.9808i	−0.539513	+	0.934463i	−0.998930	+	0.0462427i	$0.985275\pi$
$774$		0			0
$775$	3.00000			0.107763
$776$		0			0
$777$	5.00000	+	8.66025i	0.179374	+	0.310685i
$778$		0			0
$779$	32.0000			1.14652
$780$	14.0000	+	3.46410i	0.501280	+	0.124035i
$781$	6.00000			0.214697
$782$		0			0
$783$	−4.00000	−	6.92820i	−0.142948	−	0.247594i
$784$	12.0000	−	20.7846i	0.428571	−	0.742307i
$785$	46.0000			1.64181
$786$		0			0
$787$	25.5000	−	44.1673i	0.908977	−	1.57439i	0.0934886	−	0.995620i	$-0.470198\pi$
$787$	25.5000	−	44.1673i	0.908977	−	1.57439i	0.815488	−	0.578774i	$-0.196469\pi$
$788$	−40.0000			−1.42494
$789$	−3.00000	+	5.19615i	−0.106803	+	0.184988i
$790$		0			0
$791$	−8.00000	−	13.8564i	−0.284447	−	0.492677i
$792$		0			0
$793$	−32.5000	+	33.7750i	−1.15411	+	1.19939i
$794$		0			0
$795$	−10.0000	−	17.3205i	−0.354663	−	0.614295i
$796$	−7.00000	−	12.1244i	−0.248108	−	0.429736i
$797$	−13.0000	+	22.5167i	−0.460484	+	0.797581i	−0.998985	−	0.0450436i	$-0.985657\pi$
$797$	−13.0000	+	22.5167i	−0.460484	+	0.797581i	0.538501	+	0.842625i	$0.318991\pi$
$798$		0			0
$799$	−8.00000	+	13.8564i	−0.283020	+	0.490204i
$800$		0			0
$801$	8.00000			0.282666
$802$		0			0
$803$	−6.50000	−	11.2583i	−0.229380	−	0.397298i
$804$	−11.0000	−	19.0526i	−0.387940	−	0.671932i
$805$	8.00000			0.281963
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000	−	10.3923i	−0.210949	−	0.365374i	0.741063	−	0.671436i	$-0.234322\pi$
$809$	−6.00000	−	10.3923i	−0.210949	−	0.365374i	−0.952012	+	0.306062i	$0.900989\pi$
$810$		0			0
$811$	−45.0000			−1.58016			−0.790082	−	0.613001i	$-0.789962\pi$
$811$	−45.0000			−1.58016			−0.790082	+	0.613001i	$0.789962\pi$
$812$	−8.00000	+	13.8564i	−0.280745	+	0.486265i
$813$	−8.50000	+	14.7224i	−0.298108	+	0.516338i
$814$		0			0
$815$	1.00000	−	1.73205i	0.0350285	−	0.0606711i
$816$	4.00000	+	6.92820i	0.140028	+	0.242536i
$817$	−14.0000	−	24.2487i	−0.489798	−	0.848355i
$818$		0			0
$819$	−1.00000	−	3.46410i	−0.0349428	−	0.121046i
$820$	32.0000			1.11749
$821$	−27.0000	−	46.7654i	−0.942306	−	1.63212i	−0.761056	−	0.648686i	$-0.775319\pi$
$821$	−27.0000	−	46.7654i	−0.942306	−	1.63212i	−0.181250	−	0.983437i	$-0.558014\pi$
$822$		0			0
$823$	−14.0000	+	24.2487i	−0.488009	+	0.845257i	−0.999905	−	0.0137907i	$-0.995610\pi$
$823$	−14.0000	+	24.2487i	−0.488009	+	0.845257i	0.511896	+	0.859048i	$0.328943\pi$
$824$		0			0
$825$	0.500000	−	0.866025i	0.0174078	−	0.0301511i
$826$		0			0
$827$	26.0000			0.904109			0.452054	−	0.891990i	$-0.350691\pi$
$827$	26.0000			0.904109			0.452054	+	0.891990i	$0.350691\pi$
$828$	4.00000	−	6.92820i	0.139010	−	0.240772i
$829$	−14.5000	−	25.1147i	−0.503606	−	0.872271i	−0.999991	−	0.00416865i	$-0.998673\pi$
$829$	−14.5000	−	25.1147i	−0.503606	−	0.872271i	0.496385	−	0.868102i	$-0.334660\pi$
$830$		0			0
$831$	−2.00000			−0.0693792
$832$	−28.0000	−	6.92820i	−0.970725	−	0.240192i
$833$	−12.0000			−0.415775
$834$		0			0
$835$	14.0000	+	24.2487i	0.484490	+	0.839161i
$836$	4.00000	−	6.92820i	0.138343	−	0.239617i
$837$	3.00000			0.103695
$838$		0			0
$839$	−25.0000	+	43.3013i	−0.863096	+	1.49493i	0.00582968	+	0.999983i	$0.498144\pi$
$839$	−25.0000	+	43.3013i	−0.863096	+	1.49493i	−0.868926	+	0.494943i	$0.835189\pi$
$840$		0			0
$841$	−17.5000	+	30.3109i	−0.603448	+	1.04520i
$842$		0			0
$843$	12.0000	+	20.7846i	0.413302	+	0.715860i
$844$	30.0000			1.03264
$845$	23.0000	+	12.1244i	0.791224	+	0.417091i
$846$		0			0
$847$	−0.500000	−	0.866025i	−0.0171802	−	0.0297570i
$848$	20.0000	+	34.6410i	0.686803	+	1.18958i
$849$	−1.50000	+	2.59808i	−0.0514799	+	0.0891657i
$850$		0			0
$851$	−20.0000	+	34.6410i	−0.685591	+	1.18748i
$852$	6.00000	−	10.3923i	0.205557	−	0.356034i
$853$	−11.0000			−0.376633			−0.188316	−	0.982108i	$-0.560303\pi$
$853$	−11.0000			−0.376633			−0.188316	+	0.982108i	$0.560303\pi$
$854$		0			0
$855$	4.00000	+	6.92820i	0.136797	+	0.236940i
$856$		0			0
$857$	−26.0000			−0.888143			−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000			−0.888143			−0.444072	+	0.895991i	$0.646466\pi$
$858$		0			0
$859$	31.0000			1.05771			0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000			1.05771			0.528853	+	0.848713i	$0.322622\pi$
$860$	−14.0000	−	24.2487i	−0.477396	−	0.826874i
$861$	−4.00000	−	6.92820i	−0.136320	−	0.236113i
$862$		0			0
$863$	−30.0000			−1.02121			−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000			−1.02121			−0.510606	+	0.859815i	$0.670579\pi$
$864$		0			0
$865$	−10.0000	+	17.3205i	−0.340010	+	0.588915i
$866$		0			0
$867$	−6.50000	+	11.2583i	−0.220752	+	0.382353i
$868$	−3.00000	−	5.19615i	−0.101827	−	0.176369i
$869$	−3.50000	−	6.06218i	−0.118729	−	0.205645i
$870$		0			0
$871$	−11.0000	−	38.1051i	−0.372721	−	1.29114i
$872$		0			0
$873$	3.50000	+	6.06218i	0.118457	+	0.205174i
$874$		0			0
$875$	6.00000	−	10.3923i	0.202837	−	0.351324i
$876$	−26.0000			−0.878459
$877$	−19.0000	+	32.9090i	−0.641584	+	1.11126i	0.343495	+	0.939155i	$0.388389\pi$
$877$	−19.0000	+	32.9090i	−0.641584	+	1.11126i	−0.985079	+	0.172102i	$0.944944\pi$
$878$		0			0
$879$	16.0000			0.539667
$880$	4.00000	−	6.92820i	0.134840	−	0.233550i
$881$	3.00000	+	5.19615i	0.101073	+	0.175063i	0.912127	−	0.409908i	$-0.134439\pi$
$881$	3.00000	+	5.19615i	0.101073	+	0.175063i	−0.811054	+	0.584971i	$0.801106\pi$
$882$		0			0
$883$	13.0000			0.437485			0.218742	−	0.975783i	$-0.429805\pi$
$883$	13.0000			0.437485			0.218742	+	0.975783i	$0.429805\pi$
$884$	4.00000	+	13.8564i	0.134535	+	0.466041i
$885$	−16.0000			−0.537834
$886$		0			0
$887$	−18.0000	−	31.1769i	−0.604381	−	1.04682i	−0.992149	−	0.125061i	$-0.960087\pi$
$887$	−18.0000	−	31.1769i	−0.604381	−	1.04682i	0.387768	−	0.921757i	$-0.373246\pi$
$888$		0			0
$889$	1.00000			0.0335389
$890$		0			0
$891$	0.500000	−	0.866025i	0.0167506	−	0.0290129i
$892$	8.00000			0.267860
$893$	16.0000	−	27.7128i	0.535420	−	0.927374i
$894$		0			0
$895$	24.0000	+	41.5692i	0.802232	+	1.38951i
$896$		0			0
$897$	10.0000	−	10.3923i	0.333890	−	0.346989i
$898$		0			0
$899$	−12.0000	−	20.7846i	−0.400222	−	0.693206i
$900$	−1.00000	−	1.73205i	−0.0333333	−	0.0577350i
$901$	10.0000	−	17.3205i	0.333148	−	0.577030i
$902$		0			0
$903$	−3.50000	+	6.06218i	−0.116473	+	0.201737i
$904$		0			0
$905$	−4.00000			−0.132964
$906$		0			0
$907$	−14.0000	−	24.2487i	−0.464862	−	0.805165i	0.534333	−	0.845274i	$-0.320563\pi$
$907$	−14.0000	−	24.2487i	−0.464862	−	0.805165i	−0.999195	+	0.0401089i	$0.987230\pi$
$908$	−24.0000	−	41.5692i	−0.796468	−	1.37952i
$909$	−6.00000			−0.199007
$910$		0			0
$911$	28.0000			0.927681			0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000			0.927681			0.463841	+	0.885919i	$0.346471\pi$
$912$	−8.00000	−	13.8564i	−0.264906	−	0.458831i
$913$	−3.00000	−	5.19615i	−0.0992855	−	0.171968i
$914$		0			0
$915$	−26.0000			−0.859533
$916$	6.00000	−	10.3923i	0.198246	−	0.343371i
$917$	9.00000	−	15.5885i	0.297206	−	0.514776i
$918$		0			0
$919$	−26.0000	+	45.0333i	−0.857661	+	1.48551i	0.0164935	+	0.999864i	$0.494750\pi$
$919$	−26.0000	+	45.0333i	−0.857661	+	1.48551i	−0.874154	+	0.485648i	$0.838584\pi$
$920$		0			0
$921$	4.50000	+	7.79423i	0.148280	+	0.256829i
$922$		0			0
$923$	15.0000	−	15.5885i	0.493731	−	0.513100i
$924$	−2.00000			−0.0657952
$925$	5.00000	+	8.66025i	0.164399	+	0.284747i
$926$		0			0
$927$	−5.50000	+	9.52628i	−0.180644	+	0.312884i
$928$		0			0
$929$	−24.0000	+	41.5692i	−0.787414	+	1.36384i	0.140132	+	0.990133i	$0.455247\pi$
$929$	−24.0000	+	41.5692i	−0.787414	+	1.36384i	−0.927546	+	0.373709i	$0.878086\pi$
$930$		0			0
$931$	24.0000			0.786568
$932$	8.00000	−	13.8564i	0.262049	−	0.453882i
$933$	−8.00000	−	13.8564i	−0.261908	−	0.453638i
$934$		0			0
$935$	−4.00000			−0.130814
$936$		0			0
$937$	−54.0000			−1.76410			−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000			−1.76410			−0.882052	+	0.471153i	$0.843838\pi$
$938$		0			0
$939$	−4.50000	−	7.79423i	−0.146852	−	0.254355i
$940$	16.0000	−	27.7128i	0.521862	−	0.903892i
$941$	−18.0000			−0.586783			−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000			−0.586783			−0.293392	+	0.955992i	$0.594784\pi$
$942$		0			0
$943$	16.0000	−	27.7128i	0.521032	−	0.902453i
$944$	32.0000			1.04151
$945$	1.00000	−	1.73205i	0.0325300	−	0.0563436i
$946$		0			0
$947$	18.0000	+	31.1769i	0.584921	+	1.01311i	0.994885	+	0.101012i	$0.0322080\pi$
$947$	18.0000	+	31.1769i	0.584921	+	1.01311i	−0.409964	+	0.912102i	$0.634459\pi$
$948$	−14.0000			−0.454699
$949$	−45.5000	−	11.2583i	−1.47699	−	0.365461i
$950$		0			0
$951$	7.00000	+	12.1244i	0.226991	+	0.393159i
$952$		0			0
$953$	−12.0000	+	20.7846i	−0.388718	+	0.673280i	−0.992277	−	0.124039i	$-0.960415\pi$
$953$	−12.0000	+	20.7846i	−0.388718	+	0.673280i	0.603559	+	0.797318i	$0.293749\pi$
$954$		0			0
$955$		0			0
$956$	−18.0000	+	31.1769i	−0.582162	+	1.00833i
$957$	−8.00000			−0.258603
$958$		0			0
$959$	1.00000	+	1.73205i	0.0322917	+	0.0559308i
$960$	−8.00000	−	13.8564i	−0.258199	−	0.447214i
$961$	−22.0000			−0.709677
$962$		0			0
$963$	8.00000			0.257796
$964$	−10.0000	−	17.3205i	−0.322078	−	0.557856i
$965$	3.00000	+	5.19615i	0.0965734	+	0.167270i
$966$		0			0
$967$	16.0000			0.514525			0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000			0.514525			0.257263	+	0.966342i	$0.417179\pi$
$968$		0			0
$969$	−4.00000	+	6.92820i	−0.128499	+	0.222566i
$970$		0			0
$971$	−16.0000	+	27.7128i	−0.513464	+	0.889346i	0.486414	+	0.873729i	$0.338305\pi$
$971$	−16.0000	+	27.7128i	−0.513464	+	0.889346i	−0.999878	+	0.0156178i	$0.995028\pi$
$972$	−1.00000	−	1.73205i	−0.0320750	−	0.0555556i
$973$	4.50000	+	7.79423i	0.144263	+	0.249871i
$974$		0			0
$975$	−1.00000	−	3.46410i	−0.0320256	−	0.110940i
$976$	52.0000			1.66448
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$	4.00000	−	6.92820i	0.127841	−	0.221426i
$980$	24.0000			0.766652
$981$	2.50000	−	4.33013i	0.0798189	−	0.138250i
$982$		0			0
$983$	−46.0000			−1.46717			−0.733586	−	0.679597i	$-0.762155\pi$
$983$	−46.0000			−1.46717			−0.733586	+	0.679597i	$0.762155\pi$
$984$		0			0
$985$	−20.0000	−	34.6410i	−0.637253	−	1.10375i
$986$		0			0
$987$	−8.00000			−0.254643
$988$	−8.00000	−	27.7128i	−0.254514	−	0.881662i
$989$	−28.0000			−0.890348
$990$		0			0
$991$	14.0000	+	24.2487i	0.444725	+	0.770286i	0.998033	−	0.0626908i	$-0.0199682\pi$
$991$	14.0000	+	24.2487i	0.444725	+	0.770286i	−0.553308	+	0.832977i	$0.686635\pi$
$992$		0			0
$993$	23.0000			0.729883
$994$		0			0
$995$	7.00000	−	12.1244i	0.221915	−	0.384368i
$996$	−12.0000			−0.380235
$997$	1.50000	−	2.59808i	0.0475055	−	0.0822819i	−0.841295	−	0.540576i	$-0.818206\pi$
$997$	1.50000	−	2.59808i	0.0475055	−	0.0822819i	0.888800	+	0.458295i	$0.151540\pi$
$998$		0			0
$999$	5.00000	+	8.66025i	0.158193	+	0.273998i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.i.a.133.1	yes	2
13.3	even	3		5577.2.a.e.1.1		1
13.9	even	3	inner	429.2.i.a.100.1	✓	2
13.10	even	6		5577.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.i.a.100.1	✓	2	13.9	even	3	inner
429.2.i.a.133.1	yes	2	1.1	even	1	trivial
5577.2.a.b.1.1		1	13.10	even	6
5577.2.a.e.1.1		1	13.3	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:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +2.00000 q^{5} +(-0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +2.00000 q^{5} +(-0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(0.500000 + 0.866025i) q^{11} +2.00000 q^{12} +(3.50000 + 0.866025i) q^{13} +(1.00000 + 1.73205i) q^{15} +(-2.00000 - 3.46410i) q^{16} +(-1.00000 + 1.73205i) q^{17} +(2.00000 - 3.46410i) q^{19} +(2.00000 - 3.46410i) q^{20} -1.00000 q^{21} +(-2.00000 - 3.46410i) q^{23} -1.00000 q^{25} -1.00000 q^{27} +(1.00000 + 1.73205i) q^{28} +(4.00000 + 6.92820i) q^{29} -3.00000 q^{31} +(-0.500000 + 0.866025i) q^{33} +(-1.00000 + 1.73205i) q^{35} +(1.00000 + 1.73205i) q^{36} +(-5.00000 - 8.66025i) q^{37} +(1.00000 + 3.46410i) q^{39} +(4.00000 + 6.92820i) q^{41} +(3.50000 - 6.06218i) q^{43} +2.00000 q^{44} +(-1.00000 + 1.73205i) q^{45} +8.00000 q^{47} +(2.00000 - 3.46410i) q^{48} +(3.00000 + 5.19615i) q^{49} -2.00000 q^{51} +(5.00000 - 5.19615i) q^{52} -10.0000 q^{53} +(1.00000 + 1.73205i) q^{55} +4.00000 q^{57} +(-4.00000 + 6.92820i) q^{59} +4.00000 q^{60} +(-6.50000 + 11.2583i) q^{61} +(-0.500000 - 0.866025i) q^{63} -8.00000 q^{64} +(7.00000 + 1.73205i) q^{65} +(-5.50000 - 9.52628i) q^{67} +(2.00000 + 3.46410i) q^{68} +(2.00000 - 3.46410i) q^{69} +(3.00000 - 5.19615i) q^{71} -13.0000 q^{73} +(-0.500000 - 0.866025i) q^{75} +(-4.00000 - 6.92820i) q^{76} -1.00000 q^{77} -7.00000 q^{79} +(-4.00000 - 6.92820i) q^{80} +(-0.500000 - 0.866025i) q^{81} -6.00000 q^{83} +(-1.00000 + 1.73205i) q^{84} +(-2.00000 + 3.46410i) q^{85} +(-4.00000 + 6.92820i) q^{87} +(-4.00000 - 6.92820i) q^{89} +(-2.50000 + 2.59808i) q^{91} -8.00000 q^{92} +(-1.50000 - 2.59808i) q^{93} +(4.00000 - 6.92820i) q^{95} +(3.50000 - 6.06218i) q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{4} + 4 q^{5} - q^{7} - q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^4 + 4 * q^5 - q^7 - q^9	\( 2 q + q^{3} + 2 q^{4} + 4 q^{5} - q^{7} - q^{9} + q^{11} + 4 q^{12} + 7 q^{13} + 2 q^{15} - 4 q^{16} - 2 q^{17} + 4 q^{19} + 4 q^{20} - 2 q^{21} - 4 q^{23} - 2 q^{25} - 2 q^{27} + 2 q^{28} + 8 q^{29} - 6 q^{31} - q^{33} - 2 q^{35} + 2 q^{36} - 10 q^{37} + 2 q^{39} + 8 q^{41} + 7 q^{43} + 4 q^{44} - 2 q^{45} + 16 q^{47} + 4 q^{48} + 6 q^{49} - 4 q^{51} + 10 q^{52} - 20 q^{53} + 2 q^{55} + 8 q^{57} - 8 q^{59} + 8 q^{60} - 13 q^{61} - q^{63} - 16 q^{64} + 14 q^{65} - 11 q^{67} + 4 q^{68} + 4 q^{69} + 6 q^{71} - 26 q^{73} - q^{75} - 8 q^{76} - 2 q^{77} - 14 q^{79} - 8 q^{80} - q^{81} - 12 q^{83} - 2 q^{84} - 4 q^{85} - 8 q^{87} - 8 q^{89} - 5 q^{91} - 16 q^{92} - 3 q^{93} + 8 q^{95} + 7 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^4 + 4 * q^5 - q^7 - q^9 + q^11 + 4 * q^12 + 7 * q^13 + 2 * q^15 - 4 * q^16 - 2 * q^17 + 4 * q^19 + 4 * q^20 - 2 * q^21 - 4 * q^23 - 2 * q^25 - 2 * q^27 + 2 * q^28 + 8 * q^29 - 6 * q^31 - q^33 - 2 * q^35 + 2 * q^36 - 10 * q^37 + 2 * q^39 + 8 * q^41 + 7 * q^43 + 4 * q^44 - 2 * q^45 + 16 * q^47 + 4 * q^48 + 6 * q^49 - 4 * q^51 + 10 * q^52 - 20 * q^53 + 2 * q^55 + 8 * q^57 - 8 * q^59 + 8 * q^60 - 13 * q^61 - q^63 - 16 * q^64 + 14 * q^65 - 11 * q^67 + 4 * q^68 + 4 * q^69 + 6 * q^71 - 26 * q^73 - q^75 - 8 * q^76 - 2 * q^77 - 14 * q^79 - 8 * q^80 - q^81 - 12 * q^83 - 2 * q^84 - 4 * q^85 - 8 * q^87 - 8 * q^89 - 5 * q^91 - 16 * q^92 - 3 * q^93 + 8 * q^95 + 7 * q^97 - 2 * q^99

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