LMFDB - Embedded newform 28.2.f.a.19.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,2,Mod(3,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("28.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	28.f (of order $6$, 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:	$0.223581125660$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			19.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	28.19
Dual form			28.2.f.a.3.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.366025 + 1.36603i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(1.73205 - 1.73205i) q^{6} +(1.73205 + 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})$	\(q+(0.366025 + 1.36603i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(1.73205 - 1.73205i) q^{6} +(1.73205 + 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(0.633975 - 2.36603i) q^{10} +(-0.866025 + 0.500000i) q^{11} +(3.00000 + 1.73205i) q^{12} +3.46410i q^{13} +(-2.09808 + 3.09808i) q^{14} +3.00000i q^{15} +(2.00000 - 3.46410i) q^{16} +(-1.50000 + 0.866025i) q^{17} +(2.59808 - 4.50000i) q^{19} +3.46410 q^{20} +(1.50000 - 4.33013i) q^{21} +(-1.00000 - 1.00000i) q^{22} +(-0.866025 - 0.500000i) q^{23} +(-1.26795 + 4.73205i) q^{24} +(-1.00000 - 1.73205i) q^{25} +(-4.73205 + 1.26795i) q^{26} -5.19615 q^{27} +(-5.00000 - 1.73205i) q^{28} +4.00000 q^{29} +(-4.09808 + 1.09808i) q^{30} +(0.866025 + 1.50000i) q^{31} +(5.46410 + 1.46410i) q^{32} +(1.50000 + 0.866025i) q^{33} +(-1.73205 - 1.73205i) q^{34} +(-0.866025 - 4.50000i) q^{35} +(-1.50000 + 2.59808i) q^{37} +(7.09808 + 1.90192i) q^{38} +(5.19615 - 3.00000i) q^{39} +(1.26795 + 4.73205i) q^{40} -3.46410i q^{41} +(6.46410 + 0.464102i) q^{42} +2.00000i q^{43} +(1.00000 - 1.73205i) q^{44} +(0.366025 - 1.36603i) q^{46} +(-4.33013 + 7.50000i) q^{47} -6.92820 q^{48} +(-1.00000 + 6.92820i) q^{49} +(2.00000 - 2.00000i) q^{50} +(2.59808 + 1.50000i) q^{51} +(-3.46410 - 6.00000i) q^{52} +(0.500000 + 0.866025i) q^{53} +(-1.90192 - 7.09808i) q^{54} +1.73205 q^{55} +(0.535898 - 7.46410i) q^{56} -9.00000 q^{57} +(1.46410 + 5.46410i) q^{58} +(2.59808 + 4.50000i) q^{59} +(-3.00000 - 5.19615i) q^{60} +(-4.50000 - 2.59808i) q^{61} +(-1.73205 + 1.73205i) q^{62} +8.00000i q^{64} +(3.00000 - 5.19615i) q^{65} +(-0.633975 + 2.36603i) q^{66} +(-2.59808 + 1.50000i) q^{67} +(1.73205 - 3.00000i) q^{68} +1.73205i q^{69} +(5.83013 - 2.83013i) q^{70} -14.0000i q^{71} +(7.50000 - 4.33013i) q^{73} +(-4.09808 - 1.09808i) q^{74} +(-1.73205 + 3.00000i) q^{75} +10.3923i q^{76} +(-2.50000 - 0.866025i) q^{77} +(6.00000 + 6.00000i) q^{78} +(-7.79423 - 4.50000i) q^{79} +(-6.00000 + 3.46410i) q^{80} +(4.50000 + 7.79423i) q^{81} +(4.73205 - 1.26795i) q^{82} +13.8564 q^{83} +(1.73205 + 9.00000i) q^{84} +3.00000 q^{85} +(-2.73205 + 0.732051i) q^{86} +(-3.46410 - 6.00000i) q^{87} +(2.73205 + 0.732051i) q^{88} +(13.5000 + 7.79423i) q^{89} +(-6.92820 + 6.00000i) q^{91} +2.00000 q^{92} +(1.50000 - 2.59808i) q^{93} +(-11.8301 - 3.16987i) q^{94} +(-7.79423 + 4.50000i) q^{95} +(-2.53590 - 9.46410i) q^{96} -17.3205i q^{97} +(-9.83013 + 1.16987i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 6 * q^5 - 8 * q^8	$ 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8} + 6 q^{10} + 12 q^{12} + 2 q^{14} + 8 q^{16} - 6 q^{17} + 6 q^{21} - 4 q^{22} - 12 q^{24} - 4 q^{25} - 12 q^{26} - 20 q^{28} + 16 q^{29} - 6 q^{30} + 8 q^{32} + 6 q^{33} - 6 q^{37} + 18 q^{38} + 12 q^{40} + 12 q^{42} + 4 q^{44} - 2 q^{46} - 4 q^{49} + 8 q^{50} + 2 q^{53} - 18 q^{54} + 16 q^{56} - 36 q^{57} - 8 q^{58} - 12 q^{60} - 18 q^{61} + 12 q^{65} - 6 q^{66} + 6 q^{70} + 30 q^{73} - 6 q^{74} - 10 q^{77} + 24 q^{78} - 24 q^{80} + 18 q^{81} + 12 q^{82} + 12 q^{85} - 4 q^{86} + 4 q^{88} + 54 q^{89} + 8 q^{92} + 6 q^{93} - 30 q^{94} - 24 q^{96} - 22 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 6 * q^5 - 8 * q^8 + 6 * q^10 + 12 * q^12 + 2 * q^14 + 8 * q^16 - 6 * q^17 + 6 * q^21 - 4 * q^22 - 12 * q^24 - 4 * q^25 - 12 * q^26 - 20 * q^28 + 16 * q^29 - 6 * q^30 + 8 * q^32 + 6 * q^33 - 6 * q^37 + 18 * q^38 + 12 * q^40 + 12 * q^42 + 4 * q^44 - 2 * q^46 - 4 * q^49 + 8 * q^50 + 2 * q^53 - 18 * q^54 + 16 * q^56 - 36 * q^57 - 8 * q^58 - 12 * q^60 - 18 * q^61 + 12 * q^65 - 6 * q^66 + 6 * q^70 + 30 * q^73 - 6 * q^74 - 10 * q^77 + 24 * q^78 - 24 * q^80 + 18 * q^81 + 12 * q^82 + 12 * q^85 - 4 * q^86 + 4 * q^88 + 54 * q^89 + 8 * q^92 + 6 * q^93 - 30 * q^94 - 24 * q^96 - 22 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/28\mathbb{Z}\right)^\times$.

$n$	$15$	$17$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	0.366025	+	1.36603i	0.258819	+	0.965926i
$2$	0.366025	+	1.36603i	0.258819	+	0.965926i
$3$	−0.866025	−	1.50000i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$3$	−0.866025	−	1.50000i	−0.500000	−	0.866025i	−1.00000			$\pi$
$4$	−1.73205	+	1.00000i	−0.866025	+	0.500000i
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	0.125567	−	0.992085i	$-0.459925\pi$
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	−0.796387	+	0.604787i	$0.793258\pi$
$6$	1.73205	−	1.73205i	0.707107	−	0.707107i
$7$	1.73205	+	2.00000i	0.654654	+	0.755929i
$7$	1.73205	+	2.00000i	0.654654	+	0.755929i
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		0			0
$10$	0.633975	−	2.36603i	0.200480	−	0.748203i
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i	−0.624844	−	0.780750i	$-0.714837\pi$
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i	0.363727	+	0.931505i	$0.381504\pi$
$12$	3.00000	+	1.73205i	0.866025	+	0.500000i
$13$			3.46410i			0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$			3.46410i			0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$	−2.09808	+	3.09808i	−0.560734	+	0.827996i
$15$			3.00000i			0.774597i
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	−1.50000	+	0.866025i	−0.363803	+	0.210042i	−0.670748	−	0.741685i	$-0.734027\pi$
$17$	−1.50000	+	0.866025i	−0.363803	+	0.210042i	0.306944	+	0.951727i	$0.400693\pi$
$18$		0			0
$19$	2.59808	−	4.50000i	0.596040	−	1.03237i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	2.59808	−	4.50000i	0.596040	−	1.03237i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$	3.46410			0.774597
$21$	1.50000	−	4.33013i	0.327327	−	0.944911i
$22$	−1.00000	−	1.00000i	−0.213201	−	0.213201i
$23$	−0.866025	−	0.500000i	−0.180579	−	0.104257i	0.406986	−	0.913434i	$-0.366580\pi$
$23$	−0.866025	−	0.500000i	−0.180579	−	0.104257i	−0.587565	+	0.809177i	$0.699913\pi$
$24$	−1.26795	+	4.73205i	−0.258819	+	0.965926i
$25$	−1.00000	−	1.73205i	−0.200000	−	0.346410i
$26$	−4.73205	+	1.26795i	−0.928032	+	0.248665i
$27$	−5.19615			−1.00000
$28$	−5.00000	−	1.73205i	−0.944911	−	0.327327i
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$	−4.09808	+	1.09808i	−0.748203	+	0.200480i
$31$	0.866025	+	1.50000i	0.155543	+	0.269408i	0.933257	−	0.359211i	$-0.116954\pi$
$31$	0.866025	+	1.50000i	0.155543	+	0.269408i	−0.777714	+	0.628619i	$0.783621\pi$
$32$	5.46410	+	1.46410i	0.965926	+	0.258819i
$33$	1.50000	+	0.866025i	0.261116	+	0.150756i
$34$	−1.73205	−	1.73205i	−0.297044	−	0.297044i
$35$	−0.866025	−	4.50000i	−0.146385	−	0.760639i
$36$		0			0
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	−0.962580	−	0.270998i	$-0.912646\pi$
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	0.715981	+	0.698119i	$0.245980\pi$
$38$	7.09808	+	1.90192i	1.15146	+	0.308533i
$39$	5.19615	−	3.00000i	0.832050	−	0.480384i
$40$	1.26795	+	4.73205i	0.200480	+	0.748203i
$41$		−	3.46410i		−	0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$		−	3.46410i		−	0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$	6.46410	+	0.464102i	0.997433	+	0.0716124i
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	1.00000	−	1.73205i	0.150756	−	0.261116i
$45$		0			0
$46$	0.366025	−	1.36603i	0.0539675	−	0.201409i
$47$	−4.33013	+	7.50000i	−0.631614	+	1.09399i	0.355608	+	0.934635i	$0.384274\pi$
$47$	−4.33013	+	7.50000i	−0.631614	+	1.09399i	−0.987222	+	0.159352i	$0.949059\pi$
$48$	−6.92820			−1.00000
$49$	−1.00000	+	6.92820i	−0.142857	+	0.989743i
$50$	2.00000	−	2.00000i	0.282843	−	0.282843i
$51$	2.59808	+	1.50000i	0.363803	+	0.210042i
$52$	−3.46410	−	6.00000i	−0.480384	−	0.832050i
$53$	0.500000	+	0.866025i	0.0686803	+	0.118958i	0.898321	−	0.439340i	$-0.144788\pi$
$53$	0.500000	+	0.866025i	0.0686803	+	0.118958i	−0.829640	+	0.558298i	$0.811454\pi$
$54$	−1.90192	−	7.09808i	−0.258819	−	0.965926i
$55$	1.73205			0.233550
$56$	0.535898	−	7.46410i	0.0716124	−	0.997433i
$57$	−9.00000			−1.19208
$58$	1.46410	+	5.46410i	0.192246	+	0.717472i
$59$	2.59808	+	4.50000i	0.338241	+	0.585850i	0.984102	−	0.177605i	$-0.0568349\pi$
$59$	2.59808	+	4.50000i	0.338241	+	0.585850i	−0.645861	+	0.763455i	$0.723502\pi$
$60$	−3.00000	−	5.19615i	−0.387298	−	0.670820i
$61$	−4.50000	−	2.59808i	−0.576166	−	0.332650i	0.183442	−	0.983030i	$-0.441276\pi$
$61$	−4.50000	−	2.59808i	−0.576166	−	0.332650i	−0.759608	+	0.650381i	$0.774609\pi$
$62$	−1.73205	+	1.73205i	−0.219971	+	0.219971i
$63$		0			0
$64$			8.00000i			1.00000i
$65$	3.00000	−	5.19615i	0.372104	−	0.644503i
$66$	−0.633975	+	2.36603i	−0.0780369	+	0.291238i
$67$	−2.59808	+	1.50000i	−0.317406	+	0.183254i	−0.650236	−	0.759733i	$-0.725330\pi$
$67$	−2.59808	+	1.50000i	−0.317406	+	0.183254i	0.332830	+	0.942987i	$0.391996\pi$
$68$	1.73205	−	3.00000i	0.210042	−	0.363803i
$69$			1.73205i			0.208514i
$70$	5.83013	−	2.83013i	0.696833	−	0.338265i
$71$		−	14.0000i		−	1.66149i	−0.556650	−	0.830747i	$-0.687914\pi$
$71$		−	14.0000i		−	1.66149i	0.556650	−	0.830747i	$-0.312086\pi$
$72$		0			0
$73$	7.50000	−	4.33013i	0.877809	−	0.506803i	0.00787336	−	0.999969i	$-0.497494\pi$
$73$	7.50000	−	4.33013i	0.877809	−	0.506803i	0.869935	+	0.493166i	$0.164160\pi$
$74$	−4.09808	−	1.09808i	−0.476392	−	0.127649i
$75$	−1.73205	+	3.00000i	−0.200000	+	0.346410i
$76$			10.3923i			1.19208i
$77$	−2.50000	−	0.866025i	−0.284901	−	0.0986928i
$78$	6.00000	+	6.00000i	0.679366	+	0.679366i
$79$	−7.79423	−	4.50000i	−0.876919	−	0.506290i	−0.00727784	−	0.999974i	$-0.502317\pi$
$79$	−7.79423	−	4.50000i	−0.876919	−	0.506290i	−0.869641	+	0.493684i	$0.835650\pi$
$80$	−6.00000	+	3.46410i	−0.670820	+	0.387298i
$81$	4.50000	+	7.79423i	0.500000	+	0.866025i
$82$	4.73205	−	1.26795i	0.522568	−	0.140022i
$83$	13.8564			1.52094			0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564			1.52094			0.760469	+	0.649374i	$0.224969\pi$
$84$	1.73205	+	9.00000i	0.188982	+	0.981981i
$85$	3.00000			0.325396
$86$	−2.73205	+	0.732051i	−0.294605	+	0.0789391i
$87$	−3.46410	−	6.00000i	−0.371391	−	0.643268i
$88$	2.73205	+	0.732051i	0.291238	+	0.0780369i
$89$	13.5000	+	7.79423i	1.43100	+	0.826187i	0.997197	−	0.0748225i	$-0.0238390\pi$
$89$	13.5000	+	7.79423i	1.43100	+	0.826187i	0.433800	+	0.901009i	$0.357172\pi$
$90$		0			0
$91$	−6.92820	+	6.00000i	−0.726273	+	0.628971i
$92$	2.00000			0.208514
$93$	1.50000	−	2.59808i	0.155543	−	0.269408i
$94$	−11.8301	−	3.16987i	−1.22018	−	0.326947i
$95$	−7.79423	+	4.50000i	−0.799671	+	0.461690i
$96$	−2.53590	−	9.46410i	−0.258819	−	0.965926i
$97$		−	17.3205i		−	1.75863i	−0.476240	−	0.879316i	$-0.658000\pi$
$97$		−	17.3205i		−	1.75863i	0.476240	−	0.879316i	$-0.342000\pi$
$98$	−9.83013	+	1.16987i	−0.992993	+	0.118175i
$99$		0			0
$100$	3.46410	+	2.00000i	0.346410	+	0.200000i
$101$	−7.50000	+	4.33013i	−0.746278	+	0.430864i	−0.824347	−	0.566084i	$-0.808458\pi$
$101$	−7.50000	+	4.33013i	−0.746278	+	0.430864i	0.0780696	+	0.996948i	$0.475124\pi$
$102$	−1.09808	+	4.09808i	−0.108726	+	0.405770i
$103$	4.33013	−	7.50000i	0.426660	−	0.738997i	−0.569914	−	0.821705i	$-0.693023\pi$
$103$	4.33013	−	7.50000i	0.426660	−	0.738997i	0.996574	+	0.0827075i	$0.0263567\pi$
$104$	6.92820	−	6.92820i	0.679366	−	0.679366i
$105$	−6.00000	+	5.19615i	−0.585540	+	0.507093i
$106$	−1.00000	+	1.00000i	−0.0971286	+	0.0971286i
$107$	11.2583	+	6.50000i	1.08838	+	0.628379i	0.933146	−	0.359498i	$-0.117052\pi$
$107$	11.2583	+	6.50000i	1.08838	+	0.628379i	0.155238	+	0.987877i	$0.450386\pi$
$108$	9.00000	−	5.19615i	0.866025	−	0.500000i
$109$	−4.50000	−	7.79423i	−0.431022	−	0.746552i	0.565940	−	0.824447i	$-0.308513\pi$
$109$	−4.50000	−	7.79423i	−0.431022	−	0.746552i	−0.996962	+	0.0778949i	$0.975180\pi$
$110$	0.633975	+	2.36603i	0.0604471	+	0.225592i
$111$	5.19615			0.493197
$112$	10.3923	−	2.00000i	0.981981	−	0.188982i
$113$	−16.0000			−1.50515			−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000			−1.50515			−0.752577	+	0.658505i	$0.771189\pi$
$114$	−3.29423	−	12.2942i	−0.308533	−	1.15146i
$115$	0.866025	+	1.50000i	0.0807573	+	0.139876i
$116$	−6.92820	+	4.00000i	−0.643268	+	0.371391i
$117$		0			0
$118$	−5.19615	+	5.19615i	−0.478345	+	0.478345i
$119$	−4.33013	−	1.50000i	−0.396942	−	0.137505i
$120$	6.00000	−	6.00000i	0.547723	−	0.547723i
$121$	−5.00000	+	8.66025i	−0.454545	+	0.787296i
$122$	1.90192	−	7.09808i	0.172192	−	0.642630i
$123$	−5.19615	+	3.00000i	−0.468521	+	0.270501i
$124$	−3.00000	−	1.73205i	−0.269408	−	0.155543i
$125$			12.1244i			1.08444i
$126$		0			0
$127$			6.00000i			0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$			6.00000i			0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	−10.9282	+	2.92820i	−0.965926	+	0.258819i
$129$	3.00000	−	1.73205i	0.264135	−	0.152499i
$130$	8.19615	+	2.19615i	0.718850	+	0.192615i
$131$	−2.59808	+	4.50000i	−0.226995	+	0.393167i	−0.956916	−	0.290365i	$-0.906223\pi$
$131$	−2.59808	+	4.50000i	−0.226995	+	0.393167i	0.729921	+	0.683531i	$0.239557\pi$
$132$	−3.46410			−0.301511
$133$	13.5000	−	2.59808i	1.17060	−	0.225282i
$134$	−3.00000	−	3.00000i	−0.259161	−	0.259161i
$135$	7.79423	+	4.50000i	0.670820	+	0.387298i
$136$	4.73205	+	1.26795i	0.405770	+	0.108726i
$137$	−0.500000	−	0.866025i	−0.0427179	−	0.0739895i	0.843876	−	0.536538i	$-0.180268\pi$
$137$	−0.500000	−	0.866025i	−0.0427179	−	0.0739895i	−0.886594	+	0.462549i	$0.846935\pi$
$138$	−2.36603	+	0.633975i	−0.201409	+	0.0539675i
$139$	−6.92820			−0.587643			−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820			−0.587643			−0.293821	+	0.955860i	$0.594927\pi$
$140$	6.00000	+	6.92820i	0.507093	+	0.585540i
$141$	15.0000			1.26323
$142$	19.1244	−	5.12436i	1.60488	−	0.430026i
$143$	−1.73205	−	3.00000i	−0.144841	−	0.250873i
$144$		0			0
$145$	−6.00000	−	3.46410i	−0.498273	−	0.287678i
$146$	8.66025	+	8.66025i	0.716728	+	0.716728i
$147$	11.2583	−	4.50000i	0.928571	−	0.371154i
$148$		−	6.00000i		−	0.493197i
$149$	−0.500000	+	0.866025i	−0.0409616	+	0.0709476i	−0.885779	−	0.464107i	$-0.846375\pi$
$149$	−0.500000	+	0.866025i	−0.0409616	+	0.0709476i	0.844818	+	0.535054i	$0.179709\pi$
$150$	−4.73205	−	1.26795i	−0.386370	−	0.103528i
$151$	6.06218	−	3.50000i	0.493333	−	0.284826i	−0.232623	−	0.972567i	$-0.574731\pi$
$151$	6.06218	−	3.50000i	0.493333	−	0.284826i	0.725956	+	0.687741i	$0.241398\pi$
$152$	−14.1962	+	3.80385i	−1.15146	+	0.308533i
$153$		0			0
$154$	0.267949	−	3.73205i	0.0215920	−	0.300737i
$155$		−	3.00000i		−	0.240966i
$156$	−6.00000	+	10.3923i	−0.480384	+	0.832050i
$157$	1.50000	−	0.866025i	0.119713	−	0.0691164i	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	1.50000	−	0.866025i	0.119713	−	0.0691164i	0.558661	+	0.829396i	$0.311315\pi$
$158$	3.29423	−	12.2942i	0.262075	−	0.978076i
$159$	0.866025	−	1.50000i	0.0686803	−	0.118958i
$160$	−6.92820	−	6.92820i	−0.547723	−	0.547723i
$161$	−0.500000	−	2.59808i	−0.0394055	−	0.204757i
$162$	−9.00000	+	9.00000i	−0.707107	+	0.707107i
$163$	−18.1865	−	10.5000i	−1.42448	−	0.822423i	−0.427802	−	0.903873i	$-0.640712\pi$
$163$	−18.1865	−	10.5000i	−1.42448	−	0.822423i	−0.996678	+	0.0814491i	$0.974045\pi$
$164$	3.46410	+	6.00000i	0.270501	+	0.468521i
$165$	−1.50000	−	2.59808i	−0.116775	−	0.202260i
$166$	5.07180	+	18.9282i	0.393648	+	1.46911i
$167$	−17.3205			−1.34030			−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205			−1.34030			−0.670151	+	0.742225i	$0.733770\pi$
$168$	−11.6603	+	5.66025i	−0.899608	+	0.436698i
$169$	1.00000			0.0769231
$170$	1.09808	+	4.09808i	0.0842186	+	0.314308i
$171$		0			0
$172$	−2.00000	−	3.46410i	−0.152499	−	0.264135i
$173$	−10.5000	−	6.06218i	−0.798300	−	0.460899i	0.0445762	−	0.999006i	$-0.485806\pi$
$173$	−10.5000	−	6.06218i	−0.798300	−	0.460899i	−0.842876	+	0.538107i	$0.819140\pi$
$174$	6.92820	−	6.92820i	0.525226	−	0.525226i
$175$	1.73205	−	5.00000i	0.130931	−	0.377964i
$176$			4.00000i			0.301511i
$177$	4.50000	−	7.79423i	0.338241	−	0.585850i
$178$	−5.70577	+	21.2942i	−0.427666	+	1.59607i
$179$	16.4545	−	9.50000i	1.22987	−	0.710063i	0.262864	−	0.964833i	$-0.415333\pi$
$179$	16.4545	−	9.50000i	1.22987	−	0.710063i	0.967002	+	0.254770i	$0.0819996\pi$
$180$		0			0
$181$			6.92820i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$			6.92820i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$	−10.7321	−	7.26795i	−0.795513	−	0.538736i
$183$			9.00000i			0.665299i
$184$	0.732051	+	2.73205i	0.0539675	+	0.201409i
$185$	4.50000	−	2.59808i	0.330847	−	0.191014i
$186$	4.09808	+	1.09808i	0.300486	+	0.0805149i
$187$	0.866025	−	1.50000i	0.0633300	−	0.109691i
$188$		−	17.3205i		−	1.26323i
$189$	−9.00000	−	10.3923i	−0.654654	−	0.755929i
$190$	−9.00000	−	9.00000i	−0.652929	−	0.652929i
$191$	0.866025	+	0.500000i	0.0626634	+	0.0361787i	0.531004	−	0.847369i	$-0.321815\pi$
$191$	0.866025	+	0.500000i	0.0626634	+	0.0361787i	−0.468341	+	0.883548i	$0.655148\pi$
$192$	12.0000	−	6.92820i	0.866025	−	0.500000i
$193$	7.50000	+	12.9904i	0.539862	+	0.935068i	0.998911	+	0.0466572i	$0.0148568\pi$
$193$	7.50000	+	12.9904i	0.539862	+	0.935068i	−0.459049	+	0.888411i	$0.651810\pi$
$194$	23.6603	−	6.33975i	1.69871	−	0.455167i
$195$	−10.3923			−0.744208
$196$	−5.19615	−	13.0000i	−0.371154	−	0.928571i
$197$	16.0000			1.13995			0.569976	−	0.821661i	$-0.306952\pi$
$197$	16.0000			1.13995			0.569976	+	0.821661i	$0.306952\pi$
$198$		0			0
$199$	11.2583	+	19.5000i	0.798082	+	1.38232i	0.920864	+	0.389885i	$0.127485\pi$
$199$	11.2583	+	19.5000i	0.798082	+	1.38232i	−0.122782	+	0.992434i	$0.539182\pi$
$200$	−1.46410	+	5.46410i	−0.103528	+	0.386370i
$201$	4.50000	+	2.59808i	0.317406	+	0.183254i
$202$	−8.66025	−	8.66025i	−0.609333	−	0.609333i
$203$	6.92820	+	8.00000i	0.486265	+	0.561490i
$204$	−6.00000			−0.420084
$205$	−3.00000	+	5.19615i	−0.209529	+	0.362915i
$206$	11.8301	+	3.16987i	0.824244	+	0.220856i
$207$		0			0
$208$	12.0000	+	6.92820i	0.832050	+	0.480384i
$209$			5.19615i			0.359425i
$210$	−9.29423	−	6.29423i	−0.641363	−	0.434343i
$211$			10.0000i			0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$			10.0000i			0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$	−1.73205	−	1.00000i	−0.118958	−	0.0686803i
$213$	−21.0000	+	12.1244i	−1.43890	+	0.830747i
$214$	−4.75833	+	17.7583i	−0.325273	+	1.21393i
$215$	1.73205	−	3.00000i	0.118125	−	0.204598i
$216$	10.3923	+	10.3923i	0.707107	+	0.707107i
$217$	−1.50000	+	4.33013i	−0.101827	+	0.293948i
$218$	9.00000	−	9.00000i	0.609557	−	0.609557i
$219$	−12.9904	−	7.50000i	−0.877809	−	0.506803i
$220$	−3.00000	+	1.73205i	−0.202260	+	0.116775i
$221$	−3.00000	−	5.19615i	−0.201802	−	0.349531i
$222$	1.90192	+	7.09808i	0.127649	+	0.476392i
$223$	6.92820			0.463947			0.231973	−	0.972722i	$-0.425482\pi$
$223$	6.92820			0.463947			0.231973	+	0.972722i	$0.425482\pi$
$224$	6.53590	+	13.4641i	0.436698	+	0.899608i
$225$		0			0
$226$	−5.85641	−	21.8564i	−0.389562	−	1.45387i
$227$	−9.52628	−	16.5000i	−0.632281	−	1.09514i	−0.987084	−	0.160202i	$-0.948785\pi$
$227$	−9.52628	−	16.5000i	−0.632281	−	1.09514i	0.354803	−	0.934941i	$-0.384548\pi$
$228$	15.5885	−	9.00000i	1.03237	−	0.596040i
$229$	−13.5000	−	7.79423i	−0.892105	−	0.515057i	−0.0174746	−	0.999847i	$-0.505563\pi$
$229$	−13.5000	−	7.79423i	−0.892105	−	0.515057i	−0.874630	+	0.484790i	$0.838896\pi$
$230$	−1.73205	+	1.73205i	−0.114208	+	0.114208i
$231$	0.866025	+	4.50000i	0.0569803	+	0.296078i
$232$	−8.00000	−	8.00000i	−0.525226	−	0.525226i
$233$	3.50000	−	6.06218i	0.229293	−	0.397146i	−0.728306	−	0.685252i	$-0.759692\pi$
$233$	3.50000	−	6.06218i	0.229293	−	0.397146i	0.957599	+	0.288106i	$0.0930254\pi$
$234$		0			0
$235$	12.9904	−	7.50000i	0.847399	−	0.489246i
$236$	−9.00000	−	5.19615i	−0.585850	−	0.338241i
$237$			15.5885i			1.01258i
$238$	0.464102	−	6.46410i	0.0300832	−	0.419005i
$239$			20.0000i			1.29369i	0.762620	+	0.646846i	$0.223912\pi$
$239$			20.0000i			1.29369i	−0.762620	+	0.646846i	$0.776088\pi$
$240$	10.3923	+	6.00000i	0.670820	+	0.387298i
$241$	−4.50000	+	2.59808i	−0.289870	+	0.167357i	−0.637883	−	0.770133i	$-0.720190\pi$
$241$	−4.50000	+	2.59808i	−0.289870	+	0.167357i	0.348013	+	0.937490i	$0.386857\pi$
$242$	−13.6603	−	3.66025i	−0.878114	−	0.235290i
$243$		0			0
$244$	10.3923			0.665299
$245$	7.50000	−	9.52628i	0.479157	−	0.608612i
$246$	−6.00000	−	6.00000i	−0.382546	−	0.382546i
$247$	15.5885	+	9.00000i	0.991870	+	0.572656i
$248$	1.26795	−	4.73205i	0.0805149	−	0.300486i
$249$	−12.0000	−	20.7846i	−0.760469	−	1.31717i
$250$	−16.5622	+	4.43782i	−1.04748	+	0.280673i
$251$	−3.46410			−0.218652			−0.109326	−	0.994006i	$-0.534869\pi$
$251$	−3.46410			−0.218652			−0.109326	+	0.994006i	$0.534869\pi$
$252$		0			0
$253$	1.00000			0.0628695
$254$	−8.19615	+	2.19615i	−0.514272	+	0.137799i
$255$	−2.59808	−	4.50000i	−0.162698	−	0.281801i
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	4.50000	+	2.59808i	0.280702	+	0.162064i	0.633741	−	0.773545i	$-0.281518\pi$
$257$	4.50000	+	2.59808i	0.280702	+	0.162064i	−0.353039	+	0.935609i	$0.614852\pi$
$258$	3.46410	+	3.46410i	0.215666	+	0.215666i
$259$	−7.79423	+	1.50000i	−0.484310	+	0.0932055i
$260$			12.0000i			0.744208i
$261$		0			0
$262$	−7.09808	−	1.90192i	−0.438521	−	0.117501i
$263$	−19.9186	+	11.5000i	−1.22823	+	0.709120i	−0.966660	−	0.256063i	$-0.917574\pi$
$263$	−19.9186	+	11.5000i	−1.22823	+	0.709120i	−0.261573	+	0.965184i	$0.584241\pi$
$264$	−1.26795	−	4.73205i	−0.0780369	−	0.291238i
$265$		−	1.73205i		−	0.106399i
$266$	8.49038	+	17.4904i	0.520579	+	1.07240i
$267$		−	27.0000i		−	1.65237i
$268$	3.00000	−	5.19615i	0.183254	−	0.317406i
$269$	19.5000	−	11.2583i	1.18894	−	0.686433i	0.230871	−	0.972984i	$-0.425842\pi$
$269$	19.5000	−	11.2583i	1.18894	−	0.686433i	0.958065	+	0.286552i	$0.0925091\pi$
$270$	−3.29423	+	12.2942i	−0.200480	+	0.748203i
$271$	7.79423	−	13.5000i	0.473466	−	0.820067i	−0.526073	−	0.850439i	$-0.676336\pi$
$271$	7.79423	−	13.5000i	0.473466	−	0.820067i	0.999539	+	0.0303728i	$0.00966946\pi$
$272$			6.92820i			0.420084i
$273$	15.0000	+	5.19615i	0.907841	+	0.314485i
$274$	1.00000	−	1.00000i	0.0604122	−	0.0604122i
$275$	1.73205	+	1.00000i	0.104447	+	0.0603023i
$276$	−1.73205	−	3.00000i	−0.104257	−	0.180579i
$277$	6.50000	+	11.2583i	0.390547	+	0.676448i	0.992522	−	0.122068i	$-0.0389525\pi$
$277$	6.50000	+	11.2583i	0.390547	+	0.676448i	−0.601975	+	0.798515i	$0.705619\pi$
$278$	−2.53590	−	9.46410i	−0.152093	−	0.567619i
$279$		0			0
$280$	−7.26795	+	10.7321i	−0.434343	+	0.641363i
$281$	−4.00000			−0.238620			−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000			−0.238620			−0.119310	+	0.992857i	$0.538068\pi$
$282$	5.49038	+	20.4904i	0.326947	+	1.22018i
$283$	−6.06218	−	10.5000i	−0.360359	−	0.624160i	0.627661	−	0.778487i	$-0.284012\pi$
$283$	−6.06218	−	10.5000i	−0.360359	−	0.624160i	−0.988020	+	0.154327i	$0.950679\pi$
$284$	14.0000	+	24.2487i	0.830747	+	1.43890i
$285$	13.5000	+	7.79423i	0.799671	+	0.461690i
$286$	3.46410	−	3.46410i	0.204837	−	0.204837i
$287$	6.92820	−	6.00000i	0.408959	−	0.354169i
$288$		0			0
$289$	−7.00000	+	12.1244i	−0.411765	+	0.713197i
$290$	2.53590	−	9.46410i	0.148913	−	0.555751i
$291$	−25.9808	+	15.0000i	−1.52302	+	0.879316i
$292$	−8.66025	+	15.0000i	−0.506803	+	0.877809i
$293$		−	20.7846i		−	1.21425i	−0.794606	−	0.607125i	$-0.792323\pi$
$293$		−	20.7846i		−	1.21425i	0.794606	−	0.607125i	$-0.207677\pi$
$294$	10.2679	+	13.7321i	0.598839	+	0.800869i
$295$		−	9.00000i		−	0.524000i
$296$	8.19615	−	2.19615i	0.476392	−	0.127649i
$297$	4.50000	−	2.59808i	0.261116	−	0.150756i
$298$	−1.36603	−	0.366025i	−0.0791317	−	0.0212033i
$299$	1.73205	−	3.00000i	0.100167	−	0.173494i
$300$		−	6.92820i		−	0.400000i
$301$	−4.00000	+	3.46410i	−0.230556	+	0.199667i
$302$	7.00000	+	7.00000i	0.402805	+	0.402805i
$303$	12.9904	+	7.50000i	0.746278	+	0.430864i
$304$	−10.3923	−	18.0000i	−0.596040	−	1.03237i
$305$	4.50000	+	7.79423i	0.257669	+	0.446296i
$306$		0			0
$307$	20.7846			1.18624			0.593120	−	0.805114i	$-0.297896\pi$
$307$	20.7846			1.18624			0.593120	+	0.805114i	$0.297896\pi$
$308$	5.19615	−	1.00000i	0.296078	−	0.0569803i
$309$	−15.0000			−0.853320
$310$	4.09808	−	1.09808i	0.232755	−	0.0623665i
$311$	−4.33013	−	7.50000i	−0.245539	−	0.425286i	0.716744	−	0.697336i	$-0.245632\pi$
$311$	−4.33013	−	7.50000i	−0.245539	−	0.425286i	−0.962283	+	0.272050i	$0.912298\pi$
$312$	−16.3923	−	4.39230i	−0.928032	−	0.248665i
$313$	1.50000	+	0.866025i	0.0847850	+	0.0489506i	0.541793	−	0.840512i	$-0.317746\pi$
$313$	1.50000	+	0.866025i	0.0847850	+	0.0489506i	−0.457008	+	0.889463i	$0.651079\pi$
$314$	1.73205	+	1.73205i	0.0977453	+	0.0977453i
$315$		0			0
$316$	18.0000			1.01258
$317$	−5.50000	+	9.52628i	−0.308911	+	0.535049i	−0.978124	−	0.208021i	$-0.933298\pi$
$317$	−5.50000	+	9.52628i	−0.308911	+	0.535049i	0.669214	+	0.743070i	$0.266631\pi$
$318$	2.36603	+	0.633975i	0.132680	+	0.0355515i
$319$	−3.46410	+	2.00000i	−0.193952	+	0.111979i
$320$	6.92820	−	12.0000i	0.387298	−	0.670820i
$321$		−	22.5167i		−	1.25676i
$322$	3.36603	−	1.63397i	0.187581	−	0.0910578i
$323$			9.00000i			0.500773i
$324$	−15.5885	−	9.00000i	−0.866025	−	0.500000i
$325$	6.00000	−	3.46410i	0.332820	−	0.192154i
$326$	7.68653	−	28.6865i	0.425718	−	1.58880i
$327$	−7.79423	+	13.5000i	−0.431022	+	0.746552i
$328$	−6.92820	+	6.92820i	−0.382546	+	0.382546i
$329$	−22.5000	+	4.33013i	−1.24047	+	0.238728i
$330$	3.00000	−	3.00000i	0.165145	−	0.165145i
$331$	6.06218	+	3.50000i	0.333207	+	0.192377i	0.657264	−	0.753660i	$-0.271714\pi$
$331$	6.06218	+	3.50000i	0.333207	+	0.192377i	−0.324057	+	0.946038i	$0.605047\pi$
$332$	−24.0000	+	13.8564i	−1.31717	+	0.760469i
$333$		0			0
$334$	−6.33975	−	23.6603i	−0.346895	−	1.29463i
$335$	5.19615			0.283896
$336$	−12.0000	−	13.8564i	−0.654654	−	0.755929i
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	0.366025	+	1.36603i	0.0199092	+	0.0743020i
$339$	13.8564	+	24.0000i	0.752577	+	1.30350i
$340$	−5.19615	+	3.00000i	−0.281801	+	0.162698i
$341$	−1.50000	−	0.866025i	−0.0812296	−	0.0468979i
$342$		0			0
$343$	−15.5885	+	10.0000i	−0.841698	+	0.539949i
$344$	4.00000	−	4.00000i	0.215666	−	0.215666i
$345$	1.50000	−	2.59808i	0.0807573	−	0.139876i
$346$	4.43782	−	16.5622i	0.238579	−	0.890388i
$347$	11.2583	−	6.50000i	0.604379	−	0.348938i	−0.166383	−	0.986061i	$-0.553209\pi$
$347$	11.2583	−	6.50000i	0.604379	−	0.348938i	0.770762	+	0.637123i	$0.219876\pi$
$348$	12.0000	+	6.92820i	0.643268	+	0.371391i
$349$			10.3923i			0.556287i	0.960539	+	0.278144i	$0.0897191\pi$
$349$			10.3923i			0.556287i	−0.960539	+	0.278144i	$0.910281\pi$
$350$	7.46410	+	0.535898i	0.398973	+	0.0286450i
$351$		−	18.0000i		−	0.960769i
$352$	−5.46410	+	1.46410i	−0.291238	+	0.0780369i
$353$	−25.5000	+	14.7224i	−1.35723	+	0.783596i	−0.989249	−	0.146238i	$-0.953283\pi$
$353$	−25.5000	+	14.7224i	−1.35723	+	0.783596i	−0.367979	+	0.929834i	$0.619950\pi$
$354$	12.2942	+	3.29423i	0.653431	+	0.175086i
$355$	−12.1244	+	21.0000i	−0.643494	+	1.11456i
$356$	−31.1769			−1.65237
$357$	1.50000	+	7.79423i	0.0793884	+	0.412514i
$358$	19.0000	+	19.0000i	1.00418	+	1.00418i
$359$	−19.9186	−	11.5000i	−1.05126	−	0.606947i	−0.128260	−	0.991741i	$-0.540939\pi$
$359$	−19.9186	−	11.5000i	−1.05126	−	0.606947i	−0.923003	+	0.384794i	$0.874273\pi$
$360$		0			0
$361$	−4.00000	−	6.92820i	−0.210526	−	0.364642i
$362$	−9.46410	+	2.53590i	−0.497422	+	0.133284i
$363$	17.3205			0.909091
$364$	6.00000	−	17.3205i	0.314485	−	0.907841i
$365$	−15.0000			−0.785136
$366$	−12.2942	+	3.29423i	−0.642630	+	0.172192i
$367$	−0.866025	−	1.50000i	−0.0452062	−	0.0782994i	0.842537	−	0.538639i	$-0.181061\pi$
$367$	−0.866025	−	1.50000i	−0.0452062	−	0.0782994i	−0.887743	+	0.460339i	$0.847728\pi$
$368$	−3.46410	+	2.00000i	−0.180579	+	0.104257i
$369$		0			0
$370$	5.19615	+	5.19615i	0.270135	+	0.270135i
$371$	−0.866025	+	2.50000i	−0.0449618	+	0.129794i
$372$			6.00000i			0.311086i
$373$	14.5000	−	25.1147i	0.750782	−	1.30039i	−0.196663	−	0.980471i	$-0.563010\pi$
$373$	14.5000	−	25.1147i	0.750782	−	1.30039i	0.947444	−	0.319921i	$-0.103656\pi$
$374$	2.36603	+	0.633975i	0.122344	+	0.0327820i
$375$	18.1865	−	10.5000i	0.939149	−	0.542218i
$376$	23.6603	−	6.33975i	1.22018	−	0.326947i
$377$			13.8564i			0.713641i
$378$	10.9019	−	16.0981i	0.560734	−	0.827996i
$379$		−	8.00000i		−	0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$		−	8.00000i		−	0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$	9.00000	−	15.5885i	0.461690	−	0.799671i
$381$	9.00000	−	5.19615i	0.461084	−	0.266207i
$382$	−0.366025	+	1.36603i	−0.0187275	+	0.0698919i
$383$	2.59808	−	4.50000i	0.132755	−	0.229939i	−0.791982	−	0.610544i	$-0.790951\pi$
$383$	2.59808	−	4.50000i	0.132755	−	0.229939i	0.924738	+	0.380605i	$0.124284\pi$
$384$	13.8564	+	13.8564i	0.707107	+	0.707107i
$385$	3.00000	+	3.46410i	0.152894	+	0.176547i
$386$	−15.0000	+	15.0000i	−0.763480	+	0.763480i
$387$		0			0
$388$	17.3205	+	30.0000i	0.879316	+	1.52302i
$389$	9.50000	+	16.4545i	0.481669	+	0.834275i	0.999779	−	0.0210389i	$-0.00669738\pi$
$389$	9.50000	+	16.4545i	0.481669	+	0.834275i	−0.518110	+	0.855314i	$0.673364\pi$
$390$	−3.80385	−	14.1962i	−0.192615	−	0.718850i
$391$	1.73205			0.0875936
$392$	15.8564	−	11.8564i	0.800869	−	0.598839i
$393$	9.00000			0.453990
$394$	5.85641	+	21.8564i	0.295041	+	1.10111i
$395$	7.79423	+	13.5000i	0.392170	+	0.679259i
$396$		0			0
$397$	16.5000	+	9.52628i	0.828111	+	0.478110i	0.853206	−	0.521575i	$-0.174655\pi$
$397$	16.5000	+	9.52628i	0.828111	+	0.478110i	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	−22.5167	+	22.5167i	−1.12866	+	1.12866i
$399$	−15.5885	−	18.0000i	−0.780399	−	0.901127i
$400$	−8.00000			−0.400000
$401$	11.5000	−	19.9186i	0.574283	−	0.994687i	−0.421837	−	0.906672i	$-0.638614\pi$
$401$	11.5000	−	19.9186i	0.574283	−	0.994687i	0.996119	−	0.0880147i	$-0.0280523\pi$
$402$	−1.90192	+	7.09808i	−0.0948593	+	0.354020i
$403$	−5.19615	+	3.00000i	−0.258839	+	0.149441i
$404$	8.66025	−	15.0000i	0.430864	−	0.746278i
$405$		−	15.5885i		−	0.774597i
$406$	−8.39230	+	12.3923i	−0.416503	+	0.615020i
$407$		−	3.00000i		−	0.148704i
$408$	−2.19615	−	8.19615i	−0.108726	−	0.405770i
$409$	22.5000	−	12.9904i	1.11255	−	0.642333i	0.173064	−	0.984911i	$-0.444633\pi$
$409$	22.5000	−	12.9904i	1.11255	−	0.642333i	0.939490	+	0.342578i	$0.111300\pi$
$410$	−8.19615	−	2.19615i	−0.404779	−	0.108460i
$411$	−0.866025	+	1.50000i	−0.0427179	+	0.0739895i
$412$			17.3205i			0.853320i
$413$	−4.50000	+	12.9904i	−0.221431	+	0.639215i
$414$		0			0
$415$	−20.7846	−	12.0000i	−1.02028	−	0.589057i
$416$	−5.07180	+	18.9282i	−0.248665	+	0.928032i
$417$	6.00000	+	10.3923i	0.293821	+	0.508913i
$418$	−7.09808	+	1.90192i	−0.347178	+	0.0930261i
$419$	−20.7846			−1.01539			−0.507697	−	0.861536i	$-0.669503\pi$
$419$	−20.7846			−1.01539			−0.507697	+	0.861536i	$0.669503\pi$
$420$	5.19615	−	15.0000i	0.253546	−	0.731925i
$421$	−20.0000			−0.974740			−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000			−0.974740			−0.487370	+	0.873195i	$0.662044\pi$
$422$	−13.6603	+	3.66025i	−0.664971	+	0.178178i
$423$		0			0
$424$	0.732051	−	2.73205i	0.0355515	−	0.132680i
$425$	3.00000	+	1.73205i	0.145521	+	0.0840168i
$426$	−24.2487	−	24.2487i	−1.17485	−	1.17485i
$427$	−2.59808	−	13.5000i	−0.125730	−	0.653311i
$428$	−26.0000			−1.25676
$429$	−3.00000	+	5.19615i	−0.144841	+	0.250873i
$430$	4.73205	+	1.26795i	0.228200	+	0.0611459i
$431$	19.9186	−	11.5000i	0.959444	−	0.553936i	0.0634424	−	0.997985i	$-0.479792\pi$
$431$	19.9186	−	11.5000i	0.959444	−	0.553936i	0.896002	+	0.444050i	$0.146459\pi$
$432$	−10.3923	+	18.0000i	−0.500000	+	0.866025i
$433$		−	10.3923i		−	0.499422i	−0.968320	−	0.249711i	$-0.919664\pi$
$433$		−	10.3923i		−	0.499422i	0.968320	−	0.249711i	$-0.0803357\pi$
$434$	−6.46410	−	0.464102i	−0.310287	−	0.0222776i
$435$			12.0000i			0.575356i
$436$	15.5885	+	9.00000i	0.746552	+	0.431022i
$437$	−4.50000	+	2.59808i	−0.215264	+	0.124283i
$438$	5.49038	−	20.4904i	0.262341	−	0.979068i
$439$	−11.2583	+	19.5000i	−0.537331	+	0.930684i	0.461716	+	0.887028i	$0.347234\pi$
$439$	−11.2583	+	19.5000i	−0.537331	+	0.930684i	−0.999047	+	0.0436563i	$0.986099\pi$
$440$	−3.46410	−	3.46410i	−0.165145	−	0.165145i
$441$		0			0
$442$	6.00000	−	6.00000i	0.285391	−	0.285391i
$443$	14.7224	+	8.50000i	0.699484	+	0.403847i	0.807155	−	0.590339i	$-0.201006\pi$
$443$	14.7224	+	8.50000i	0.699484	+	0.403847i	−0.107671	+	0.994187i	$0.534339\pi$
$444$	−9.00000	+	5.19615i	−0.427121	+	0.246598i
$445$	−13.5000	−	23.3827i	−0.639961	−	1.10845i
$446$	2.53590	+	9.46410i	0.120078	+	0.448138i
$447$	1.73205			0.0819232
$448$	−16.0000	+	13.8564i	−0.755929	+	0.654654i
$449$	8.00000			0.377543			0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000			0.377543			0.188772	+	0.982021i	$0.439549\pi$
$450$		0			0
$451$	1.73205	+	3.00000i	0.0815591	+	0.141264i
$452$	27.7128	−	16.0000i	1.30350	−	0.752577i
$453$	−10.5000	−	6.06218i	−0.493333	−	0.284826i
$454$	19.0526	−	19.0526i	0.894181	−	0.894181i
$455$	15.5885	−	3.00000i	0.730798	−	0.140642i
$456$	18.0000	+	18.0000i	0.842927	+	0.842927i
$457$	−7.50000	+	12.9904i	−0.350835	+	0.607664i	−0.986396	−	0.164386i	$-0.947436\pi$
$457$	−7.50000	+	12.9904i	−0.350835	+	0.607664i	0.635561	+	0.772051i	$0.280769\pi$
$458$	5.70577	−	21.2942i	0.266613	−	0.995014i
$459$	7.79423	−	4.50000i	0.363803	−	0.210042i
$460$	−3.00000	−	1.73205i	−0.139876	−	0.0807573i
$461$		−	17.3205i		−	0.806696i	−0.915047	−	0.403348i	$-0.867846\pi$
$461$		−	17.3205i		−	0.806696i	0.915047	−	0.403348i	$-0.132154\pi$
$462$	−5.83013	+	2.83013i	−0.271242	+	0.131669i
$463$		−	30.0000i		−	1.39422i	−0.716965	−	0.697109i	$-0.754469\pi$
$463$		−	30.0000i		−	1.39422i	0.716965	−	0.697109i	$-0.245531\pi$
$464$	8.00000	−	13.8564i	0.371391	−	0.643268i
$465$	−4.50000	+	2.59808i	−0.208683	+	0.120483i
$466$	9.56218	+	2.56218i	0.442959	+	0.118691i
$467$	4.33013	−	7.50000i	0.200374	−	0.347059i	−0.748275	−	0.663389i	$-0.769117\pi$
$467$	4.33013	−	7.50000i	0.200374	−	0.347059i	0.948649	+	0.316330i	$0.102451\pi$
$468$		0			0
$469$	−7.50000	−	2.59808i	−0.346318	−	0.119968i
$470$	15.0000	+	15.0000i	0.691898	+	0.691898i
$471$	−2.59808	−	1.50000i	−0.119713	−	0.0691164i
$472$	3.80385	−	14.1962i	0.175086	−	0.653431i
$473$	−1.00000	−	1.73205i	−0.0459800	−	0.0796398i
$474$	−21.2942	+	5.70577i	−0.978076	+	0.262075i
$475$	−10.3923			−0.476832
$476$	9.00000	−	1.73205i	0.412514	−	0.0793884i
$477$		0			0
$478$	−27.3205	+	7.32051i	−1.24961	+	0.334832i
$479$	6.06218	+	10.5000i	0.276988	+	0.479757i	0.970635	−	0.240558i	$-0.0773304\pi$
$479$	6.06218	+	10.5000i	0.276988	+	0.479757i	−0.693647	+	0.720315i	$0.743997\pi$
$480$	−4.39230	+	16.3923i	−0.200480	+	0.748203i
$481$	−9.00000	−	5.19615i	−0.410365	−	0.236924i
$482$	−5.19615	−	5.19615i	−0.236678	−	0.236678i
$483$	−3.46410	+	3.00000i	−0.157622	+	0.136505i
$484$		−	20.0000i		−	0.909091i
$485$	−15.0000	+	25.9808i	−0.681115	+	1.17973i
$486$		0			0
$487$	−26.8468	+	15.5000i	−1.21654	+	0.702372i	−0.964177	−	0.265260i	$-0.914542\pi$
$487$	−26.8468	+	15.5000i	−1.21654	+	0.702372i	−0.252367	+	0.967632i	$0.581209\pi$
$488$	3.80385	+	14.1962i	0.172192	+	0.642630i
$489$			36.3731i			1.64485i
$490$	15.7583	+	6.75833i	0.711889	+	0.305310i
$491$			32.0000i			1.44414i	0.691820	+	0.722070i	$0.256809\pi$
$491$			32.0000i			1.44414i	−0.691820	+	0.722070i	$0.743191\pi$
$492$	6.00000	−	10.3923i	0.270501	−	0.468521i
$493$	−6.00000	+	3.46410i	−0.270226	+	0.156015i
$494$	−6.58846	+	24.5885i	−0.296429	+	1.10629i
$495$		0			0
$496$	6.92820			0.311086
$497$	28.0000	−	24.2487i	1.25597	−	1.08770i
$498$	24.0000	−	24.0000i	1.07547	−	1.07547i
$499$	30.3109	+	17.5000i	1.35690	+	0.783408i	0.989205	−	0.146538i	$-0.0468131\pi$
$499$	30.3109	+	17.5000i	1.35690	+	0.783408i	0.367697	+	0.929946i	$0.380146\pi$
$500$	−12.1244	−	21.0000i	−0.542218	−	0.939149i
$501$	15.0000	+	25.9808i	0.670151	+	1.16073i
$502$	−1.26795	−	4.73205i	−0.0565913	−	0.211202i
$503$	−6.92820			−0.308913			−0.154457	−	0.988000i	$-0.549363\pi$
$503$	−6.92820			−0.308913			−0.154457	+	0.988000i	$0.549363\pi$
$504$		0			0
$505$	15.0000			0.667491
$506$	0.366025	+	1.36603i	0.0162718	+	0.0607272i
$507$	−0.866025	−	1.50000i	−0.0384615	−	0.0666173i
$508$	−6.00000	−	10.3923i	−0.266207	−	0.461084i
$509$	10.5000	+	6.06218i	0.465404	+	0.268701i	0.714314	−	0.699825i	$-0.246739\pi$
$509$	10.5000	+	6.06218i	0.465404	+	0.268701i	−0.248910	+	0.968527i	$0.580072\pi$
$510$	5.19615	−	5.19615i	0.230089	−	0.230089i
$511$	21.6506	+	7.50000i	0.957768	+	0.331780i
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$	−13.5000	+	23.3827i	−0.596040	+	1.03237i
$514$	−1.90192	+	7.09808i	−0.0838903	+	0.313083i
$515$	−12.9904	+	7.50000i	−0.572425	+	0.330489i
$516$	−3.46410	+	6.00000i	−0.152499	+	0.264135i
$517$		−	8.66025i		−	0.380878i
$518$	−4.90192	−	10.0981i	−0.215378	−	0.443684i
$519$			21.0000i			0.921798i
$520$	−16.3923	+	4.39230i	−0.718850	+	0.192615i
$521$	1.50000	−	0.866025i	0.0657162	−	0.0379413i	−0.466782	−	0.884372i	$-0.654587\pi$
$521$	1.50000	−	0.866025i	0.0657162	−	0.0379413i	0.532498	+	0.846431i	$0.321253\pi$
$522$		0			0
$523$	12.9904	−	22.5000i	0.568030	−	0.983856i	−0.428731	−	0.903432i	$-0.641039\pi$
$523$	12.9904	−	22.5000i	0.568030	−	0.983856i	0.996761	−	0.0804241i	$-0.0256275\pi$
$524$		−	10.3923i		−	0.453990i
$525$	−9.00000	+	1.73205i	−0.392792	+	0.0755929i
$526$	−23.0000	−	23.0000i	−1.00285	−	1.00285i
$527$	−2.59808	−	1.50000i	−0.113174	−	0.0653410i
$528$	6.00000	−	3.46410i	0.261116	−	0.150756i
$529$	−11.0000	−	19.0526i	−0.478261	−	0.828372i
$530$	2.36603	−	0.633975i	0.102774	−	0.0275381i
$531$		0			0
$532$	−20.7846	+	18.0000i	−0.901127	+	0.780399i
$533$	12.0000			0.519778
$534$	36.8827	−	9.88269i	1.59607	−	0.427666i
$535$	−11.2583	−	19.5000i	−0.486740	−	0.843059i
$536$	8.19615	+	2.19615i	0.354020	+	0.0948593i
$537$	−28.5000	−	16.4545i	−1.22987	−	0.710063i
$538$	22.5167	+	22.5167i	0.970762	+	0.970762i
$539$	−2.59808	−	6.50000i	−0.111907	−	0.279975i
$540$	−18.0000			−0.774597
$541$	9.50000	−	16.4545i	0.408437	−	0.707433i	−0.586278	−	0.810110i	$-0.699407\pi$
$541$	9.50000	−	16.4545i	0.408437	−	0.707433i	0.994715	+	0.102677i	$0.0327407\pi$
$542$	21.2942	+	5.70577i	0.914665	+	0.245084i
$543$	10.3923	−	6.00000i	0.445976	−	0.257485i
$544$	−9.46410	+	2.53590i	−0.405770	+	0.108726i
$545$			15.5885i			0.667736i
$546$	−1.60770	+	22.3923i	−0.0688030	+	0.958302i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	1.73205	+	1.00000i	0.0739895	+	0.0427179i
$549$		0			0
$550$	−0.732051	+	2.73205i	−0.0312148	+	0.116495i
$551$	10.3923	−	18.0000i	0.442727	−	0.766826i
$552$	3.46410	−	3.46410i	0.147442	−	0.147442i
$553$	−4.50000	−	23.3827i	−0.191359	−	0.994333i
$554$	−13.0000	+	13.0000i	−0.552317	+	0.552317i
$555$	−7.79423	−	4.50000i	−0.330847	−	0.191014i
$556$	12.0000	−	6.92820i	0.508913	−	0.293821i
$557$	−18.5000	−	32.0429i	−0.783870	−	1.35770i	−0.929672	−	0.368389i	$-0.879909\pi$
$557$	−18.5000	−	32.0429i	−0.783870	−	1.35770i	0.145802	−	0.989314i	$-0.453424\pi$
$558$		0			0
$559$	−6.92820			−0.293032
$560$	−17.3205	−	6.00000i	−0.731925	−	0.253546i
$561$	−3.00000			−0.126660
$562$	−1.46410	−	5.46410i	−0.0617594	−	0.230489i
$563$	−11.2583	−	19.5000i	−0.474482	−	0.821827i	0.525091	−	0.851046i	$-0.324031\pi$
$563$	−11.2583	−	19.5000i	−0.474482	−	0.821827i	−0.999573	+	0.0292191i	$0.990698\pi$
$564$	−25.9808	+	15.0000i	−1.09399	+	0.631614i
$565$	24.0000	+	13.8564i	1.00969	+	0.582943i
$566$	12.1244	−	12.1244i	0.509625	−	0.509625i
$567$	−7.79423	+	22.5000i	−0.327327	+	0.944911i
$568$	−28.0000	+	28.0000i	−1.17485	+	1.17485i
$569$	6.50000	−	11.2583i	0.272494	−	0.471974i	−0.697006	−	0.717066i	$-0.745485\pi$
$569$	6.50000	−	11.2583i	0.272494	−	0.471974i	0.969500	+	0.245092i	$0.0788181\pi$
$570$	−5.70577	+	21.2942i	−0.238988	+	0.891917i
$571$	−18.1865	+	10.5000i	−0.761083	+	0.439411i	−0.829684	−	0.558233i	$-0.811480\pi$
$571$	−18.1865	+	10.5000i	−0.761083	+	0.439411i	0.0686016	+	0.997644i	$0.478146\pi$
$572$	6.00000	+	3.46410i	0.250873	+	0.144841i
$573$		−	1.73205i		−	0.0723575i
$574$	10.7321	+	7.26795i	0.447947	+	0.303358i
$575$			2.00000i			0.0834058i
$576$		0			0
$577$	−28.5000	+	16.4545i	−1.18647	+	0.685009i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−28.5000	+	16.4545i	−1.18647	+	0.685009i	−0.228968	+	0.973434i	$0.573535\pi$
$578$	−19.1244	−	5.12436i	−0.795468	−	0.213145i
$579$	12.9904	−	22.5000i	0.539862	−	0.935068i
$580$	13.8564			0.575356
$581$	24.0000	+	27.7128i	0.995688	+	1.14972i
$582$	−30.0000	−	30.0000i	−1.24354	−	1.24354i
$583$	−0.866025	−	0.500000i	−0.0358671	−	0.0207079i
$584$	−23.6603	−	6.33975i	−0.979068	−	0.262341i
$585$		0			0
$586$	28.3923	−	7.60770i	1.17288	−	0.314271i
$587$	6.92820			0.285958			0.142979	−	0.989726i	$-0.454332\pi$
$587$	6.92820			0.285958			0.142979	+	0.989726i	$0.454332\pi$
$588$	−15.0000	+	19.0526i	−0.618590	+	0.785714i
$589$	9.00000			0.370839
$590$	12.2942	−	3.29423i	0.506145	−	0.135621i
$591$	−13.8564	−	24.0000i	−0.569976	−	0.987228i
$592$	6.00000	+	10.3923i	0.246598	+	0.427121i
$593$	−13.5000	−	7.79423i	−0.554379	−	0.320071i	0.196508	−	0.980502i	$-0.437040\pi$
$593$	−13.5000	−	7.79423i	−0.554379	−	0.320071i	−0.750886	+	0.660432i	$0.770373\pi$
$594$	5.19615	+	5.19615i	0.213201	+	0.213201i
$595$	5.19615	+	6.00000i	0.213021	+	0.245976i
$596$		−	2.00000i		−	0.0819232i
$597$	19.5000	−	33.7750i	0.798082	−	1.38232i
$598$	4.73205	+	1.26795i	0.193508	+	0.0518503i
$599$	14.7224	−	8.50000i	0.601542	−	0.347301i	−0.168106	−	0.985769i	$-0.553765\pi$
$599$	14.7224	−	8.50000i	0.601542	−	0.347301i	0.769648	+	0.638468i	$0.220432\pi$
$600$	9.46410	−	2.53590i	0.386370	−	0.103528i
$601$			38.1051i			1.55434i	0.629291	+	0.777170i	$0.283346\pi$
$601$			38.1051i			1.55434i	−0.629291	+	0.777170i	$0.716654\pi$
$602$	−6.19615	−	4.19615i	−0.252536	−	0.171022i
$603$		0			0
$604$	−7.00000	+	12.1244i	−0.284826	+	0.493333i
$605$	15.0000	−	8.66025i	0.609837	−	0.352089i
$606$	−5.49038	+	20.4904i	−0.223031	+	0.832365i
$607$	−7.79423	+	13.5000i	−0.316358	+	0.547948i	−0.979725	−	0.200346i	$-0.935793\pi$
$607$	−7.79423	+	13.5000i	−0.316358	+	0.547948i	0.663367	+	0.748294i	$0.269127\pi$
$608$	20.7846	−	20.7846i	0.842927	−	0.842927i
$609$	6.00000	−	17.3205i	0.243132	−	0.701862i
$610$	−9.00000	+	9.00000i	−0.364399	+	0.364399i
$611$	−25.9808	−	15.0000i	−1.05107	−	0.606835i
$612$		0			0
$613$	15.5000	+	26.8468i	0.626039	+	1.08433i	0.988339	+	0.152270i	$0.0486583\pi$
$613$	15.5000	+	26.8468i	0.626039	+	1.08433i	−0.362300	+	0.932062i	$0.618008\pi$
$614$	7.60770	+	28.3923i	0.307022	+	1.14582i
$615$	10.3923			0.419058
$616$	3.26795	+	6.73205i	0.131669	+	0.271242i
$617$	20.0000			0.805170			0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000			0.805170			0.402585	+	0.915383i	$0.368112\pi$
$618$	−5.49038	−	20.4904i	−0.220856	−	0.824244i
$619$	−7.79423	−	13.5000i	−0.313276	−	0.542611i	0.665793	−	0.746136i	$-0.268093\pi$
$619$	−7.79423	−	13.5000i	−0.313276	−	0.542611i	−0.979070	+	0.203526i	$0.934760\pi$
$620$	3.00000	+	5.19615i	0.120483	+	0.208683i
$621$	4.50000	+	2.59808i	0.180579	+	0.104257i
$622$	8.66025	−	8.66025i	0.347245	−	0.347245i
$623$	7.79423	+	40.5000i	0.312269	+	1.62260i
$624$		−	24.0000i		−	0.960769i
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$	−0.633975	+	2.36603i	−0.0253387	+	0.0945654i
$627$	7.79423	−	4.50000i	0.311272	−	0.179713i
$628$	−1.73205	+	3.00000i	−0.0691164	+	0.119713i
$629$		−	5.19615i		−	0.207184i
$630$		0			0
$631$			30.0000i			1.19428i	0.802137	+	0.597141i	$0.203697\pi$
$631$			30.0000i			1.19428i	−0.802137	+	0.597141i	$0.796303\pi$
$632$	6.58846	+	24.5885i	0.262075	+	0.978076i
$633$	15.0000	−	8.66025i	0.596196	−	0.344214i
$634$	−15.0263	−	4.02628i	−0.596770	−	0.159904i
$635$	5.19615	−	9.00000i	0.206203	−	0.357154i
$636$			3.46410i			0.137361i
$637$	−24.0000	−	3.46410i	−0.950915	−	0.137253i
$638$	−4.00000	−	4.00000i	−0.158362	−	0.158362i
$639$		0			0
$640$	18.9282	+	5.07180i	0.748203	+	0.200480i
$641$	6.50000	+	11.2583i	0.256735	+	0.444677i	0.965365	−	0.260902i	$-0.0840201\pi$
$641$	6.50000	+	11.2583i	0.256735	+	0.444677i	−0.708631	+	0.705580i	$0.750687\pi$
$642$	30.7583	−	8.24167i	1.21393	−	0.325273i
$643$	−13.8564			−0.546443			−0.273222	−	0.961951i	$-0.588089\pi$
$643$	−13.8564			−0.546443			−0.273222	+	0.961951i	$0.588089\pi$
$644$	3.46410	+	4.00000i	0.136505	+	0.157622i
$645$	−6.00000			−0.236250
$646$	−12.2942	+	3.29423i	−0.483710	+	0.129610i
$647$	16.4545	+	28.5000i	0.646892	+	1.12045i	0.983861	+	0.178935i	$0.0572651\pi$
$647$	16.4545	+	28.5000i	0.646892	+	1.12045i	−0.336968	+	0.941516i	$0.609402\pi$
$648$	6.58846	−	24.5885i	0.258819	−	0.965926i
$649$	−4.50000	−	2.59808i	−0.176640	−	0.101983i
$650$	6.92820	+	6.92820i	0.271746	+	0.271746i
$651$	7.79423	−	1.50000i	0.305480	−	0.0587896i
$652$	42.0000			1.64485
$653$	−15.5000	+	26.8468i	−0.606562	+	1.05060i	0.385241	+	0.922816i	$0.374118\pi$
$653$	−15.5000	+	26.8468i	−0.606562	+	1.05060i	−0.991803	+	0.127780i	$0.959215\pi$
$654$	−21.2942	−	5.70577i	−0.832670	−	0.223113i
$655$	7.79423	−	4.50000i	0.304546	−	0.175830i
$656$	−12.0000	−	6.92820i	−0.468521	−	0.270501i
$657$		0			0
$658$	−14.1506	−	29.1506i	−0.551649	−	1.13641i
$659$		−	38.0000i		−	1.48027i	−0.672458	−	0.740135i	$-0.734762\pi$
$659$		−	38.0000i		−	1.48027i	0.672458	−	0.740135i	$-0.265238\pi$
$660$	5.19615	+	3.00000i	0.202260	+	0.116775i
$661$	−34.5000	+	19.9186i	−1.34189	+	0.774743i	−0.987085	−	0.160196i	$-0.948788\pi$
$661$	−34.5000	+	19.9186i	−1.34189	+	0.774743i	−0.354809	+	0.934939i	$0.615454\pi$
$662$	−2.56218	+	9.56218i	−0.0995819	+	0.371645i
$663$	−5.19615	+	9.00000i	−0.201802	+	0.349531i
$664$	−27.7128	−	27.7128i	−1.07547	−	1.07547i
$665$	−22.5000	−	7.79423i	−0.872513	−	0.302247i
$666$		0			0
$667$	−3.46410	−	2.00000i	−0.134131	−	0.0774403i
$668$	30.0000	−	17.3205i	1.16073	−	0.670151i
$669$	−6.00000	−	10.3923i	−0.231973	−	0.401790i
$670$	1.90192	+	7.09808i	0.0734777	+	0.274223i
$671$	5.19615			0.200595
$672$	14.5359	−	21.4641i	0.560734	−	0.827996i
$673$	24.0000			0.925132			0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000			0.925132			0.462566	+	0.886585i	$0.346929\pi$
$674$		0			0
$675$	5.19615	+	9.00000i	0.200000	+	0.346410i
$676$	−1.73205	+	1.00000i	−0.0666173	+	0.0384615i
$677$	−37.5000	−	21.6506i	−1.44124	−	0.832102i	−0.443309	−	0.896369i	$-0.646196\pi$
$677$	−37.5000	−	21.6506i	−1.44124	−	0.832102i	−0.997933	+	0.0642672i	$0.979529\pi$
$678$	−27.7128	+	27.7128i	−1.06430	+	1.06430i
$679$	34.6410	−	30.0000i	1.32940	−	1.15129i
$680$	−6.00000	−	6.00000i	−0.230089	−	0.230089i
$681$	−16.5000	+	28.5788i	−0.632281	+	1.09514i
$682$	0.633975	−	2.36603i	0.0242761	−	0.0905998i
$683$	−21.6506	+	12.5000i	−0.828439	+	0.478299i	−0.853318	−	0.521391i	$-0.825413\pi$
$683$	−21.6506	+	12.5000i	−0.828439	+	0.478299i	0.0248792	+	0.999690i	$0.492080\pi$
$684$		0			0
$685$			1.73205i			0.0661783i
$686$	−19.3660	−	17.6340i	−0.739398	−	0.673268i
$687$			27.0000i			1.03011i
$688$	6.92820	+	4.00000i	0.264135	+	0.152499i
$689$	−3.00000	+	1.73205i	−0.114291	+	0.0659859i
$690$	4.09808	+	1.09808i	0.156011	+	0.0418030i
$691$	−6.06218	+	10.5000i	−0.230616	+	0.399439i	−0.957990	−	0.286803i	$-0.907407\pi$
$691$	−6.06218	+	10.5000i	−0.230616	+	0.399439i	0.727373	+	0.686242i	$0.240741\pi$
$692$	24.2487			0.921798
$693$		0			0
$694$	13.0000	+	13.0000i	0.493473	+	0.493473i
$695$	10.3923	+	6.00000i	0.394203	+	0.227593i
$696$	−5.07180	+	18.9282i	−0.192246	+	0.717472i
$697$	3.00000	+	5.19615i	0.113633	+	0.196818i
$698$	−14.1962	+	3.80385i	−0.537332	+	0.143978i
$699$	−12.1244			−0.458585
$700$	2.00000	+	10.3923i	0.0755929	+	0.392792i
$701$	−26.0000			−0.982006			−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000			−0.982006			−0.491003	+	0.871158i	$0.663370\pi$
$702$	24.5885	−	6.58846i	0.928032	−	0.248665i
$703$	7.79423	+	13.5000i	0.293965	+	0.509162i
$704$	−4.00000	−	6.92820i	−0.150756	−	0.261116i
$705$	−22.5000	−	12.9904i	−0.847399	−	0.489246i
$706$	−29.4449	−	29.4449i	−1.10817	−	1.10817i
$707$	−21.6506	−	7.50000i	−0.814256	−	0.282067i
$708$			18.0000i			0.676481i
$709$	4.50000	−	7.79423i	0.169001	−	0.292718i	−0.769068	−	0.639167i	$-0.779279\pi$
$709$	4.50000	−	7.79423i	0.169001	−	0.292718i	0.938069	+	0.346449i	$0.112613\pi$
$710$	−33.1244	−	8.87564i	−1.24313	−	0.333097i
$711$		0			0
$712$	−11.4115	−	42.5885i	−0.427666	−	1.59607i
$713$		−	1.73205i		−	0.0648658i
$714$	−10.0981	+	4.90192i	−0.377911	+	0.183450i
$715$			6.00000i			0.224387i
$716$	−19.0000	+	32.9090i	−0.710063	+	1.22987i
$717$	30.0000	−	17.3205i	1.12037	−	0.646846i
$718$	8.41858	−	31.4186i	0.314179	−	1.17253i
$719$	−12.9904	+	22.5000i	−0.484459	+	0.839108i	−0.999841	−	0.0178527i	$-0.994317\pi$
$719$	−12.9904	+	22.5000i	−0.484459	+	0.839108i	0.515381	+	0.856961i	$0.327650\pi$
$720$		0			0
$721$	22.5000	−	4.33013i	0.837944	−	0.161262i
$722$	8.00000	−	8.00000i	0.297729	−	0.297729i
$723$	7.79423	+	4.50000i	0.289870	+	0.167357i
$724$	−6.92820	−	12.0000i	−0.257485	−	0.445976i
$725$	−4.00000	−	6.92820i	−0.148556	−	0.257307i
$726$	6.33975	+	23.6603i	0.235290	+	0.878114i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	25.8564	+	1.85641i	0.958302	+	0.0688030i
$729$	27.0000			1.00000
$730$	−5.49038	−	20.4904i	−0.203208	−	0.758383i
$731$	−1.73205	−	3.00000i	−0.0640622	−	0.110959i
$732$	−9.00000	−	15.5885i	−0.332650	−	0.576166i
$733$	37.5000	+	21.6506i	1.38509	+	0.799684i	0.992757	−	0.120137i	$-0.0383334\pi$
$733$	37.5000	+	21.6506i	1.38509	+	0.799684i	0.392337	+	0.919822i	$0.371667\pi$
$734$	1.73205	−	1.73205i	0.0639312	−	0.0639312i
$735$	−20.7846	−	3.00000i	−0.766652	−	0.110657i
$736$	−4.00000	−	4.00000i	−0.147442	−	0.147442i
$737$	1.50000	−	2.59808i	0.0552532	−	0.0957014i
$738$		0			0
$739$	44.1673	−	25.5000i	1.62472	−	0.938033i	0.639087	−	0.769135i	$-0.279313\pi$
$739$	44.1673	−	25.5000i	1.62472	−	0.938033i	0.985634	−	0.168898i	$-0.0540208\pi$
$740$	−5.19615	+	9.00000i	−0.191014	+	0.330847i
$741$		−	31.1769i		−	1.14531i
$742$	−3.73205	−	0.267949i	−0.137008	−	0.00983672i
$743$		−	34.0000i		−	1.24734i	−0.781688	−	0.623670i	$-0.785641\pi$
$743$		−	34.0000i		−	1.24734i	0.781688	−	0.623670i	$-0.214359\pi$
$744$	−8.19615	+	2.19615i	−0.300486	+	0.0805149i
$745$	1.50000	−	0.866025i	0.0549557	−	0.0317287i
$746$	39.6147	+	10.6147i	1.45040	+	0.388633i
$747$		0			0
$748$			3.46410i			0.126660i
$749$	6.50000	+	33.7750i	0.237505	+	1.23411i
$750$	21.0000	+	21.0000i	0.766812	+	0.766812i
$751$	21.6506	+	12.5000i	0.790043	+	0.456131i	0.839978	−	0.542621i	$-0.182568\pi$
$751$	21.6506	+	12.5000i	0.790043	+	0.456131i	−0.0499348	+	0.998752i	$0.515901\pi$
$752$	17.3205	+	30.0000i	0.631614	+	1.09399i
$753$	3.00000	+	5.19615i	0.109326	+	0.189358i
$754$	−18.9282	+	5.07180i	−0.689325	+	0.184704i
$755$	−12.1244			−0.441250
$756$	25.9808	+	9.00000i	0.944911	+	0.327327i
$757$	−48.0000			−1.74459			−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000			−1.74459			−0.872295	+	0.488980i	$0.837369\pi$
$758$	10.9282	−	2.92820i	0.396930	−	0.106357i
$759$	−0.866025	−	1.50000i	−0.0314347	−	0.0544466i
$760$	24.5885	+	6.58846i	0.891917	+	0.238988i
$761$	16.5000	+	9.52628i	0.598125	+	0.345327i	0.768303	−	0.640086i	$-0.221101\pi$
$761$	16.5000	+	9.52628i	0.598125	+	0.345327i	−0.170179	+	0.985413i	$0.554435\pi$
$762$	10.3923	+	10.3923i	0.376473	+	0.376473i
$763$	7.79423	−	22.5000i	0.282170	−	0.814555i
$764$	−2.00000			−0.0723575
$765$		0			0
$766$	7.09808	+	1.90192i	0.256464	+	0.0687193i
$767$	−15.5885	+	9.00000i	−0.562867	+	0.324971i
$768$	−13.8564	+	24.0000i	−0.500000	+	0.866025i
$769$		−	3.46410i		−	0.124919i	−0.998048	−	0.0624593i	$-0.980106\pi$
$769$		−	3.46410i		−	0.124919i	0.998048	−	0.0624593i	$-0.0198944\pi$

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:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	28.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(1.73205 - 1.73205i) q^{6} +(1.73205 + 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})\)	\(q+(0.366025 + 1.36603i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(1.73205 - 1.73205i) q^{6} +(1.73205 + 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(0.633975 - 2.36603i) q^{10} +(-0.866025 + 0.500000i) q^{11} +(3.00000 + 1.73205i) q^{12} +3.46410i q^{13} +(-2.09808 + 3.09808i) q^{14} +3.00000i q^{15} +(2.00000 - 3.46410i) q^{16} +(-1.50000 + 0.866025i) q^{17} +(2.59808 - 4.50000i) q^{19} +3.46410 q^{20} +(1.50000 - 4.33013i) q^{21} +(-1.00000 - 1.00000i) q^{22} +(-0.866025 - 0.500000i) q^{23} +(-1.26795 + 4.73205i) q^{24} +(-1.00000 - 1.73205i) q^{25} +(-4.73205 + 1.26795i) q^{26} -5.19615 q^{27} +(-5.00000 - 1.73205i) q^{28} +4.00000 q^{29} +(-4.09808 + 1.09808i) q^{30} +(0.866025 + 1.50000i) q^{31} +(5.46410 + 1.46410i) q^{32} +(1.50000 + 0.866025i) q^{33} +(-1.73205 - 1.73205i) q^{34} +(-0.866025 - 4.50000i) q^{35} +(-1.50000 + 2.59808i) q^{37} +(7.09808 + 1.90192i) q^{38} +(5.19615 - 3.00000i) q^{39} +(1.26795 + 4.73205i) q^{40} -3.46410i q^{41} +(6.46410 + 0.464102i) q^{42} +2.00000i q^{43} +(1.00000 - 1.73205i) q^{44} +(0.366025 - 1.36603i) q^{46} +(-4.33013 + 7.50000i) q^{47} -6.92820 q^{48} +(-1.00000 + 6.92820i) q^{49} +(2.00000 - 2.00000i) q^{50} +(2.59808 + 1.50000i) q^{51} +(-3.46410 - 6.00000i) q^{52} +(0.500000 + 0.866025i) q^{53} +(-1.90192 - 7.09808i) q^{54} +1.73205 q^{55} +(0.535898 - 7.46410i) q^{56} -9.00000 q^{57} +(1.46410 + 5.46410i) q^{58} +(2.59808 + 4.50000i) q^{59} +(-3.00000 - 5.19615i) q^{60} +(-4.50000 - 2.59808i) q^{61} +(-1.73205 + 1.73205i) q^{62} +8.00000i q^{64} +(3.00000 - 5.19615i) q^{65} +(-0.633975 + 2.36603i) q^{66} +(-2.59808 + 1.50000i) q^{67} +(1.73205 - 3.00000i) q^{68} +1.73205i q^{69} +(5.83013 - 2.83013i) q^{70} -14.0000i q^{71} +(7.50000 - 4.33013i) q^{73} +(-4.09808 - 1.09808i) q^{74} +(-1.73205 + 3.00000i) q^{75} +10.3923i q^{76} +(-2.50000 - 0.866025i) q^{77} +(6.00000 + 6.00000i) q^{78} +(-7.79423 - 4.50000i) q^{79} +(-6.00000 + 3.46410i) q^{80} +(4.50000 + 7.79423i) q^{81} +(4.73205 - 1.26795i) q^{82} +13.8564 q^{83} +(1.73205 + 9.00000i) q^{84} +3.00000 q^{85} +(-2.73205 + 0.732051i) q^{86} +(-3.46410 - 6.00000i) q^{87} +(2.73205 + 0.732051i) q^{88} +(13.5000 + 7.79423i) q^{89} +(-6.92820 + 6.00000i) q^{91} +2.00000 q^{92} +(1.50000 - 2.59808i) q^{93} +(-11.8301 - 3.16987i) q^{94} +(-7.79423 + 4.50000i) q^{95} +(-2.53590 - 9.46410i) q^{96} -17.3205i q^{97} +(-9.83013 + 1.16987i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 6 * q^5 - 8 * q^8	\( 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8} + 6 q^{10} + 12 q^{12} + 2 q^{14} + 8 q^{16} - 6 q^{17} + 6 q^{21} - 4 q^{22} - 12 q^{24} - 4 q^{25} - 12 q^{26} - 20 q^{28} + 16 q^{29} - 6 q^{30} + 8 q^{32} + 6 q^{33} - 6 q^{37} + 18 q^{38} + 12 q^{40} + 12 q^{42} + 4 q^{44} - 2 q^{46} - 4 q^{49} + 8 q^{50} + 2 q^{53} - 18 q^{54} + 16 q^{56} - 36 q^{57} - 8 q^{58} - 12 q^{60} - 18 q^{61} + 12 q^{65} - 6 q^{66} + 6 q^{70} + 30 q^{73} - 6 q^{74} - 10 q^{77} + 24 q^{78} - 24 q^{80} + 18 q^{81} + 12 q^{82} + 12 q^{85} - 4 q^{86} + 4 q^{88} + 54 q^{89} + 8 q^{92} + 6 q^{93} - 30 q^{94} - 24 q^{96} - 22 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 6 * q^5 - 8 * q^8 + 6 * q^10 + 12 * q^12 + 2 * q^14 + 8 * q^16 - 6 * q^17 + 6 * q^21 - 4 * q^22 - 12 * q^24 - 4 * q^25 - 12 * q^26 - 20 * q^28 + 16 * q^29 - 6 * q^30 + 8 * q^32 + 6 * q^33 - 6 * q^37 + 18 * q^38 + 12 * q^40 + 12 * q^42 + 4 * q^44 - 2 * q^46 - 4 * q^49 + 8 * q^50 + 2 * q^53 - 18 * q^54 + 16 * q^56 - 36 * q^57 - 8 * q^58 - 12 * q^60 - 18 * q^61 + 12 * q^65 - 6 * q^66 + 6 * q^70 + 30 * q^73 - 6 * q^74 - 10 * q^77 + 24 * q^78 - 24 * q^80 + 18 * q^81 + 12 * q^82 + 12 * q^85 - 4 * q^86 + 4 * q^88 + 54 * q^89 + 8 * q^92 + 6 * q^93 - 30 * q^94 - 24 * q^96 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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