LMFDB - Embedded newform 81.2.c.b.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,2,Mod(28,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.646788256372$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	81.55
Dual form			81.2.c.b.28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 1.50000i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.866025 + 1.50000i) q^{5} +(-1.00000 + 1.73205i) q^{7} -1.73205 q^{8} +O(q^{10})$	\(q+(-0.866025 + 1.50000i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.866025 + 1.50000i) q^{5} +(-1.00000 + 1.73205i) q^{7} -1.73205 q^{8} -3.00000 q^{10} +(1.73205 - 3.00000i) q^{11} +(0.500000 + 0.866025i) q^{13} +(-1.73205 - 3.00000i) q^{14} +(2.50000 - 4.33013i) q^{16} +5.19615 q^{17} +2.00000 q^{19} +(0.866025 - 1.50000i) q^{20} +(3.00000 + 5.19615i) q^{22} +(-1.73205 - 3.00000i) q^{23} +(1.00000 - 1.73205i) q^{25} -1.73205 q^{26} +2.00000 q^{28} +(-0.866025 + 1.50000i) q^{29} +(-4.00000 - 6.92820i) q^{31} +(2.59808 + 4.50000i) q^{32} +(-4.50000 + 7.79423i) q^{34} -3.46410 q^{35} -7.00000 q^{37} +(-1.73205 + 3.00000i) q^{38} +(-1.50000 - 2.59808i) q^{40} +(3.46410 + 6.00000i) q^{41} +(-1.00000 + 1.73205i) q^{43} -3.46410 q^{44} +6.00000 q^{46} +(-3.46410 + 6.00000i) q^{47} +(1.50000 + 2.59808i) q^{49} +(1.73205 + 3.00000i) q^{50} +(0.500000 - 0.866025i) q^{52} +6.00000 q^{55} +(1.73205 - 3.00000i) q^{56} +(-1.50000 - 2.59808i) q^{58} +(-6.92820 - 12.0000i) q^{59} +(3.50000 - 6.06218i) q^{61} +13.8564 q^{62} +1.00000 q^{64} +(-0.866025 + 1.50000i) q^{65} +(5.00000 + 8.66025i) q^{67} +(-2.59808 - 4.50000i) q^{68} +(3.00000 - 5.19615i) q^{70} -10.3923 q^{71} -7.00000 q^{73} +(6.06218 - 10.5000i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(3.46410 + 6.00000i) q^{77} +(-1.00000 + 1.73205i) q^{79} +8.66025 q^{80} -12.0000 q^{82} +(6.92820 - 12.0000i) q^{83} +(4.50000 + 7.79423i) q^{85} +(-1.73205 - 3.00000i) q^{86} +(-3.00000 + 5.19615i) q^{88} -5.19615 q^{89} -2.00000 q^{91} +(-1.73205 + 3.00000i) q^{92} +(-6.00000 - 10.3923i) q^{94} +(1.73205 + 3.00000i) q^{95} +(-1.00000 + 1.73205i) q^{97} -5.19615 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} - 4 q^{7}+O(q^{10}) $ 4 * q - 2 * q^4 - 4 * q^7	$ 4 q - 2 q^{4} - 4 q^{7} - 12 q^{10} + 2 q^{13} + 10 q^{16} + 8 q^{19} + 12 q^{22} + 4 q^{25} + 8 q^{28} - 16 q^{31} - 18 q^{34} - 28 q^{37} - 6 q^{40} - 4 q^{43} + 24 q^{46} + 6 q^{49} + 2 q^{52} + 24 q^{55} - 6 q^{58} + 14 q^{61} + 4 q^{64} + 20 q^{67} + 12 q^{70} - 28 q^{73} - 4 q^{76} - 4 q^{79} - 48 q^{82} + 18 q^{85} - 12 q^{88} - 8 q^{91} - 24 q^{94} - 4 q^{97}+O(q^{100}) $ 4 * q - 2 * q^4 - 4 * q^7 - 12 * q^10 + 2 * q^13 + 10 * q^16 + 8 * q^19 + 12 * q^22 + 4 * q^25 + 8 * q^28 - 16 * q^31 - 18 * q^34 - 28 * q^37 - 6 * q^40 - 4 * q^43 + 24 * q^46 + 6 * q^49 + 2 * q^52 + 24 * q^55 - 6 * q^58 + 14 * q^61 + 4 * q^64 + 20 * q^67 + 12 * q^70 - 28 * q^73 - 4 * q^76 - 4 * q^79 - 48 * q^82 + 18 * q^85 - 12 * q^88 - 8 * q^91 - 24 * q^94 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.866025	+	1.50000i	−0.612372	+	1.06066i	0.378467	+	0.925615i	$0.376451\pi$
$2$	−0.866025	+	1.50000i	−0.612372	+	1.06066i	−0.990839	+	0.135045i	$0.956882\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	0.866025	+	1.50000i	0.387298	+	0.670820i	0.992085	−	0.125567i	$-0.0400750\pi$
$5$	0.866025	+	1.50000i	0.387298	+	0.670820i	−0.604787	+	0.796387i	$0.706742\pi$
$6$		0			0
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	$-0.956709\pi$
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	0.612801	+	0.790237i	$0.290043\pi$
$8$	−1.73205			−0.612372
$9$		0			0
$10$	−3.00000			−0.948683
$11$	1.73205	−	3.00000i	0.522233	−	0.904534i	−0.477432	−	0.878668i	$-0.658432\pi$
$11$	1.73205	−	3.00000i	0.522233	−	0.904534i	0.999665	−	0.0258656i	$-0.00823419\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	0.926995	−	0.375073i	$-0.122382\pi$
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	−0.788320	+	0.615265i	$0.789049\pi$
$14$	−1.73205	−	3.00000i	−0.462910	−	0.801784i
$15$		0			0
$16$	2.50000	−	4.33013i	0.625000	−	1.08253i
$17$	5.19615			1.26025			0.630126	−	0.776493i	$-0.283003\pi$
$17$	5.19615			1.26025			0.630126	+	0.776493i	$0.283003\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$	0.866025	−	1.50000i	0.193649	−	0.335410i
$21$		0			0
$22$	3.00000	+	5.19615i	0.639602	+	1.10782i
$23$	−1.73205	−	3.00000i	−0.361158	−	0.625543i	0.626994	−	0.779024i	$-0.284285\pi$
$23$	−1.73205	−	3.00000i	−0.361158	−	0.625543i	−0.988152	+	0.153481i	$0.950952\pi$
$24$		0			0
$25$	1.00000	−	1.73205i	0.200000	−	0.346410i
$26$	−1.73205			−0.339683
$27$		0			0
$28$	2.00000			0.377964
$29$	−0.866025	+	1.50000i	−0.160817	+	0.278543i	−0.935162	−	0.354221i	$-0.884746\pi$
$29$	−0.866025	+	1.50000i	−0.160817	+	0.278543i	0.774345	+	0.632764i	$0.218080\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$	2.59808	+	4.50000i	0.459279	+	0.795495i
$33$		0			0
$34$	−4.50000	+	7.79423i	−0.771744	+	1.33670i
$35$	−3.46410			−0.585540
$36$		0			0
$37$	−7.00000			−1.15079			−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000			−1.15079			−0.575396	+	0.817875i	$0.695152\pi$
$38$	−1.73205	+	3.00000i	−0.280976	+	0.486664i
$39$		0			0
$40$	−1.50000	−	2.59808i	−0.237171	−	0.410792i
$41$	3.46410	+	6.00000i	0.541002	+	0.937043i	0.998847	+	0.0480106i	$0.0152881\pi$
$41$	3.46410	+	6.00000i	0.541002	+	0.937043i	−0.457845	+	0.889032i	$0.651379\pi$
$42$		0			0
$43$	−1.00000	+	1.73205i	−0.152499	+	0.264135i	−0.932145	−	0.362084i	$-0.882065\pi$
$43$	−1.00000	+	1.73205i	−0.152499	+	0.264135i	0.779647	+	0.626219i	$0.215399\pi$
$44$	−3.46410			−0.522233
$45$		0			0
$46$	6.00000			0.884652
$47$	−3.46410	+	6.00000i	−0.505291	+	0.875190i	0.494690	+	0.869069i	$0.335282\pi$
$47$	−3.46410	+	6.00000i	−0.505291	+	0.875190i	−0.999981	+	0.00612051i	$0.998052\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$	1.73205	+	3.00000i	0.244949	+	0.424264i
$51$		0			0
$52$	0.500000	−	0.866025i	0.0693375	−	0.120096i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$	1.73205	−	3.00000i	0.231455	−	0.400892i
$57$		0			0
$58$	−1.50000	−	2.59808i	−0.196960	−	0.341144i
$59$	−6.92820	−	12.0000i	−0.901975	−	1.56227i	−0.824927	−	0.565240i	$-0.808784\pi$
$59$	−6.92820	−	12.0000i	−0.901975	−	1.56227i	−0.0770484	−	0.997027i	$-0.524550\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$	13.8564			1.75977
$63$		0			0
$64$	1.00000			0.125000
$65$	−0.866025	+	1.50000i	−0.107417	+	0.186052i
$66$		0			0
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	0.991098	+	0.133135i	$0.0425044\pi$
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	−0.380251	+	0.924883i	$0.624162\pi$
$68$	−2.59808	−	4.50000i	−0.315063	−	0.545705i
$69$		0			0
$70$	3.00000	−	5.19615i	0.358569	−	0.621059i
$71$	−10.3923			−1.23334			−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923			−1.23334			−0.616670	+	0.787222i	$0.711519\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$	6.06218	−	10.5000i	0.704714	−	1.22060i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$	3.46410	+	6.00000i	0.394771	+	0.683763i
$78$		0			0
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	−0.916781	−	0.399390i	$-0.869222\pi$
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	0.804272	+	0.594261i	$0.202555\pi$
$80$	8.66025			0.968246
$81$		0			0
$82$	−12.0000			−1.32518
$83$	6.92820	−	12.0000i	0.760469	−	1.31717i	−0.182140	−	0.983273i	$-0.558302\pi$
$83$	6.92820	−	12.0000i	0.760469	−	1.31717i	0.942609	−	0.333899i	$-0.108364\pi$
$84$		0			0
$85$	4.50000	+	7.79423i	0.488094	+	0.845403i
$86$	−1.73205	−	3.00000i	−0.186772	−	0.323498i
$87$		0			0
$88$	−3.00000	+	5.19615i	−0.319801	+	0.553912i
$89$	−5.19615			−0.550791			−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615			−0.550791			−0.275396	+	0.961331i	$0.588809\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$	−1.73205	+	3.00000i	−0.180579	+	0.312772i
$93$		0			0
$94$	−6.00000	−	10.3923i	−0.618853	−	1.07188i
$95$	1.73205	+	3.00000i	0.177705	+	0.307794i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$	−5.19615			−0.524891
$99$		0			0
$100$	−2.00000			−0.200000
$101$	−3.46410	+	6.00000i	−0.344691	+	0.597022i	−0.985298	−	0.170847i	$-0.945350\pi$
$101$	−3.46410	+	6.00000i	−0.344691	+	0.597022i	0.640607	+	0.767869i	$0.278683\pi$
$102$		0			0
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	0.598858	−	0.800855i	$-0.295621\pi$
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	−0.992990	+	0.118199i	$0.962288\pi$
$104$	−0.866025	−	1.50000i	−0.0849208	−	0.147087i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$	−5.19615	+	9.00000i	−0.495434	+	0.858116i
$111$		0			0
$112$	5.00000	+	8.66025i	0.472456	+	0.818317i
$113$	0.866025	+	1.50000i	0.0814688	+	0.141108i	0.903881	−	0.427784i	$-0.140706\pi$
$113$	0.866025	+	1.50000i	0.0814688	+	0.141108i	−0.822412	+	0.568892i	$0.807372\pi$
$114$		0			0
$115$	3.00000	−	5.19615i	0.279751	−	0.484544i
$116$	1.73205			0.160817
$117$		0			0
$118$	24.0000			2.20938
$119$	−5.19615	+	9.00000i	−0.476331	+	0.825029i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$	6.06218	+	10.5000i	0.548844	+	0.950625i
$123$		0			0
$124$	−4.00000	+	6.92820i	−0.359211	+	0.622171i
$125$	12.1244			1.08444
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$	−6.06218	+	10.5000i	−0.535826	+	0.928078i
$129$		0			0
$130$	−1.50000	−	2.59808i	−0.131559	−	0.227866i
$131$	−1.73205	−	3.00000i	−0.151330	−	0.262111i	0.780387	−	0.625297i	$-0.215022\pi$
$131$	−1.73205	−	3.00000i	−0.151330	−	0.262111i	−0.931717	+	0.363186i	$0.881689\pi$
$132$		0			0
$133$	−2.00000	+	3.46410i	−0.173422	+	0.300376i
$134$	−17.3205			−1.49626
$135$		0			0
$136$	−9.00000			−0.771744
$137$	−0.866025	+	1.50000i	−0.0739895	+	0.128154i	−0.900646	−	0.434553i	$-0.856906\pi$
$137$	−0.866025	+	1.50000i	−0.0739895	+	0.128154i	0.826657	+	0.562706i	$0.190240\pi$
$138$		0			0
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	0.645021	−	0.764165i	$-0.276849\pi$
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	−0.984297	+	0.176522i	$0.943515\pi$
$140$	1.73205	+	3.00000i	0.146385	+	0.253546i
$141$		0			0
$142$	9.00000	−	15.5885i	0.755263	−	1.30815i
$143$	3.46410			0.289683
$144$		0			0
$145$	−3.00000			−0.249136
$146$	6.06218	−	10.5000i	0.501709	−	0.868986i
$147$		0			0
$148$	3.50000	+	6.06218i	0.287698	+	0.498308i
$149$	−4.33013	−	7.50000i	−0.354738	−	0.614424i	0.632335	−	0.774695i	$-0.282097\pi$
$149$	−4.33013	−	7.50000i	−0.354738	−	0.614424i	−0.987073	+	0.160271i	$0.948763\pi$
$150$		0			0
$151$	−10.0000	+	17.3205i	−0.813788	+	1.40952i	0.0964061	+	0.995342i	$0.469265\pi$
$151$	−10.0000	+	17.3205i	−0.813788	+	1.40952i	−0.910195	+	0.414181i	$0.864068\pi$
$152$	−3.46410			−0.280976
$153$		0			0
$154$	−12.0000			−0.966988
$155$	6.92820	−	12.0000i	0.556487	−	0.963863i
$156$		0			0
$157$	−8.50000	−	14.7224i	−0.678374	−	1.17498i	−0.975470	−	0.220131i	$-0.929352\pi$
$157$	−8.50000	−	14.7224i	−0.678374	−	1.17498i	0.297097	−	0.954847i	$-0.403982\pi$
$158$	−1.73205	−	3.00000i	−0.137795	−	0.238667i
$159$		0			0
$160$	−4.50000	+	7.79423i	−0.355756	+	0.616188i
$161$	6.92820			0.546019
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$	3.46410	−	6.00000i	0.270501	−	0.468521i
$165$		0			0
$166$	12.0000	+	20.7846i	0.931381	+	1.61320i
$167$	8.66025	+	15.0000i	0.670151	+	1.16073i	0.977861	+	0.209255i	$0.0671038\pi$
$167$	8.66025	+	15.0000i	0.670151	+	1.16073i	−0.307711	+	0.951480i	$0.599563\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$	−15.5885			−1.19558
$171$		0			0
$172$	2.00000			0.152499
$173$	9.52628	−	16.5000i	0.724270	−	1.25447i	−0.235004	−	0.971994i	$-0.575510\pi$
$173$	9.52628	−	16.5000i	0.724270	−	1.25447i	0.959274	−	0.282477i	$-0.0911562\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$	−8.66025	−	15.0000i	−0.652791	−	1.13067i
$177$		0			0
$178$	4.50000	−	7.79423i	0.337289	−	0.584202i
$179$	20.7846			1.55351			0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846			1.55351			0.776757	+	0.629800i	$0.216863\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$	1.73205	−	3.00000i	0.128388	−	0.222375i
$183$		0			0
$184$	3.00000	+	5.19615i	0.221163	+	0.383065i
$185$	−6.06218	−	10.5000i	−0.445700	−	0.771975i
$186$		0			0
$187$	9.00000	−	15.5885i	0.658145	−	1.13994i
$188$	6.92820			0.505291
$189$		0			0
$190$	−6.00000			−0.435286
$191$	−8.66025	+	15.0000i	−0.626634	+	1.08536i	0.361588	+	0.932338i	$0.382235\pi$
$191$	−8.66025	+	15.0000i	−0.626634	+	1.08536i	−0.988222	+	0.153024i	$0.951099\pi$
$192$		0			0
$193$	0.500000	+	0.866025i	0.0359908	+	0.0623379i	0.883460	−	0.468507i	$-0.155208\pi$
$193$	0.500000	+	0.866025i	0.0359908	+	0.0623379i	−0.847469	+	0.530845i	$0.821875\pi$
$194$	−1.73205	−	3.00000i	−0.124354	−	0.215387i
$195$		0			0
$196$	1.50000	−	2.59808i	0.107143	−	0.185577i
$197$	−5.19615			−0.370211			−0.185105	−	0.982719i	$-0.559263\pi$
$197$	−5.19615			−0.370211			−0.185105	+	0.982719i	$0.559263\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$	−1.73205	+	3.00000i	−0.122474	+	0.212132i
$201$		0			0
$202$	−6.00000	−	10.3923i	−0.422159	−	0.731200i
$203$	−1.73205	−	3.00000i	−0.121566	−	0.210559i
$204$		0			0
$205$	−6.00000	+	10.3923i	−0.419058	+	0.725830i
$206$	13.8564			0.965422
$207$		0			0
$208$	5.00000			0.346688
$209$	3.46410	−	6.00000i	0.239617	−	0.415029i
$210$		0			0
$211$	5.00000	+	8.66025i	0.344214	+	0.596196i	0.985211	−	0.171347i	$-0.0548120\pi$
$211$	5.00000	+	8.66025i	0.344214	+	0.596196i	−0.640996	+	0.767544i	$0.721479\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.46410			−0.236250
$216$		0			0
$217$	16.0000			1.08615
$218$	−9.52628	+	16.5000i	−0.645201	+	1.11752i
$219$		0			0
$220$	−3.00000	−	5.19615i	−0.202260	−	0.350325i
$221$	2.59808	+	4.50000i	0.174766	+	0.302703i
$222$		0			0
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	−0.897564	−	0.440884i	$-0.854665\pi$
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	0.830599	+	0.556871i	$0.187998\pi$
$224$	−10.3923			−0.694365
$225$		0			0
$226$	−3.00000			−0.199557
$227$	1.73205	−	3.00000i	0.114960	−	0.199117i	−0.802804	−	0.596244i	$-0.796659\pi$
$227$	1.73205	−	3.00000i	0.114960	−	0.199117i	0.917764	+	0.397127i	$0.129993\pi$
$228$		0			0
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	0.882073	−	0.471113i	$-0.156147\pi$
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	−0.849032	+	0.528341i	$0.822814\pi$
$230$	5.19615	+	9.00000i	0.342624	+	0.593442i
$231$		0			0
$232$	1.50000	−	2.59808i	0.0984798	−	0.170572i
$233$	−25.9808			−1.70206			−0.851028	−	0.525120i	$-0.824020\pi$
$233$	−25.9808			−1.70206			−0.851028	+	0.525120i	$0.824020\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$	−6.92820	+	12.0000i	−0.450988	+	0.781133i
$237$		0			0
$238$	−9.00000	−	15.5885i	−0.583383	−	1.01045i
$239$	13.8564	+	24.0000i	0.896296	+	1.55243i	0.832192	+	0.554488i	$0.187086\pi$
$239$	13.8564	+	24.0000i	0.896296	+	1.55243i	0.0641045	+	0.997943i	$0.479581\pi$
$240$		0			0
$241$	−14.5000	+	25.1147i	−0.934027	+	1.61778i	−0.157667	+	0.987492i	$0.550397\pi$
$241$	−14.5000	+	25.1147i	−0.934027	+	1.61778i	−0.776360	+	0.630290i	$0.782936\pi$
$242$	1.73205			0.111340
$243$		0			0
$244$	−7.00000			−0.448129
$245$	−2.59808	+	4.50000i	−0.165985	+	0.287494i
$246$		0			0
$247$	1.00000	+	1.73205i	0.0636285	+	0.110208i
$248$	6.92820	+	12.0000i	0.439941	+	0.762001i
$249$		0			0
$250$	−10.5000	+	18.1865i	−0.664078	+	1.15022i
$251$	10.3923			0.655956			0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923			0.655956			0.327978	+	0.944685i	$0.393633\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$	−1.73205	+	3.00000i	−0.108679	+	0.188237i
$255$		0			0
$256$	−9.50000	−	16.4545i	−0.593750	−	1.02841i
$257$	−4.33013	−	7.50000i	−0.270106	−	0.467837i	0.698783	−	0.715334i	$-0.253725\pi$
$257$	−4.33013	−	7.50000i	−0.270106	−	0.467837i	−0.968889	+	0.247497i	$0.920392\pi$
$258$		0			0
$259$	7.00000	−	12.1244i	0.434959	−	0.753371i
$260$	1.73205			0.107417
$261$		0			0
$262$	6.00000			0.370681
$263$	−3.46410	+	6.00000i	−0.213606	+	0.369976i	−0.952840	−	0.303472i	$-0.901854\pi$
$263$	−3.46410	+	6.00000i	−0.213606	+	0.369976i	0.739235	+	0.673448i	$0.235187\pi$
$264$		0			0
$265$		0			0
$266$	−3.46410	−	6.00000i	−0.212398	−	0.367884i
$267$		0			0
$268$	5.00000	−	8.66025i	0.305424	−	0.529009i
$269$	−15.5885			−0.950445			−0.475223	−	0.879866i	$-0.657632\pi$
$269$	−15.5885			−0.950445			−0.475223	+	0.879866i	$0.657632\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$	12.9904	−	22.5000i	0.787658	−	1.36426i
$273$		0			0
$274$	−1.50000	−	2.59808i	−0.0906183	−	0.156956i
$275$	−3.46410	−	6.00000i	−0.208893	−	0.361814i
$276$		0			0
$277$	−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	$0.185804\pi$
$278$	13.8564			0.831052
$279$		0			0
$280$	6.00000			0.358569
$281$	−6.06218	+	10.5000i	−0.361639	+	0.626377i	−0.988231	−	0.152970i	$-0.951116\pi$
$281$	−6.06218	+	10.5000i	−0.361639	+	0.626377i	0.626592	+	0.779348i	$0.284449\pi$
$282$		0			0
$283$	14.0000	+	24.2487i	0.832214	+	1.44144i	0.896279	+	0.443491i	$0.146260\pi$
$283$	14.0000	+	24.2487i	0.832214	+	1.44144i	−0.0640654	+	0.997946i	$0.520407\pi$
$284$	5.19615	+	9.00000i	0.308335	+	0.534052i
$285$		0			0
$286$	−3.00000	+	5.19615i	−0.177394	+	0.307255i
$287$	−13.8564			−0.817918
$288$		0			0
$289$	10.0000			0.588235
$290$	2.59808	−	4.50000i	0.152564	−	0.264249i
$291$		0			0
$292$	3.50000	+	6.06218i	0.204822	+	0.354762i
$293$	−9.52628	−	16.5000i	−0.556531	−	0.963940i	−0.997783	−	0.0665568i	$-0.978799\pi$
$293$	−9.52628	−	16.5000i	−0.556531	−	0.963940i	0.441251	−	0.897384i	$-0.354535\pi$
$294$		0			0
$295$	12.0000	−	20.7846i	0.698667	−	1.21013i
$296$	12.1244			0.704714
$297$		0			0
$298$	15.0000			0.868927
$299$	1.73205	−	3.00000i	0.100167	−	0.173494i
$300$		0			0
$301$	−2.00000	−	3.46410i	−0.115278	−	0.199667i
$302$	−17.3205	−	30.0000i	−0.996683	−	1.72631i
$303$		0			0
$304$	5.00000	−	8.66025i	0.286770	−	0.496700i
$305$	12.1244			0.694239
$306$		0			0
$307$	−16.0000			−0.913168			−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000			−0.913168			−0.456584	+	0.889680i	$0.650927\pi$
$308$	3.46410	−	6.00000i	0.197386	−	0.341882i
$309$		0			0
$310$	12.0000	+	20.7846i	0.681554	+	1.18049i
$311$	3.46410	+	6.00000i	0.196431	+	0.340229i	0.947369	−	0.320144i	$-0.103731\pi$
$311$	3.46410	+	6.00000i	0.196431	+	0.340229i	−0.750938	+	0.660373i	$0.770398\pi$
$312$		0			0
$313$	12.5000	−	21.6506i	0.706542	−	1.22377i	−0.259590	−	0.965719i	$-0.583588\pi$
$313$	12.5000	−	21.6506i	0.706542	−	1.22377i	0.966132	−	0.258047i	$-0.0830791\pi$
$314$	29.4449			1.66167
$315$		0			0
$316$	2.00000			0.112509
$317$	4.33013	−	7.50000i	0.243204	−	0.421242i	−0.718421	−	0.695609i	$-0.755135\pi$
$317$	4.33013	−	7.50000i	0.243204	−	0.421242i	0.961625	+	0.274367i	$0.0884682\pi$
$318$		0			0
$319$	3.00000	+	5.19615i	0.167968	+	0.290929i
$320$	0.866025	+	1.50000i	0.0484123	+	0.0838525i
$321$		0			0
$322$	−6.00000	+	10.3923i	−0.334367	+	0.579141i
$323$	10.3923			0.578243
$324$		0			0
$325$	2.00000			0.110940
$326$	13.8564	−	24.0000i	0.767435	−	1.32924i
$327$		0			0
$328$	−6.00000	−	10.3923i	−0.331295	−	0.573819i
$329$	−6.92820	−	12.0000i	−0.381964	−	0.661581i
$330$		0			0
$331$	−1.00000	+	1.73205i	−0.0549650	+	0.0952021i	−0.892199	−	0.451643i	$-0.850838\pi$
$331$	−1.00000	+	1.73205i	−0.0549650	+	0.0952021i	0.837234	+	0.546845i	$0.184171\pi$
$332$	−13.8564			−0.760469
$333$		0			0
$334$	−30.0000			−1.64153
$335$	−8.66025	+	15.0000i	−0.473160	+	0.819538i
$336$		0			0
$337$	−13.0000	−	22.5167i	−0.708155	−	1.22656i	−0.965541	−	0.260252i	$-0.916194\pi$
$337$	−13.0000	−	22.5167i	−0.708155	−	1.22656i	0.257386	−	0.966309i	$-0.417139\pi$
$338$	10.3923	+	18.0000i	0.565267	+	0.979071i
$339$		0			0
$340$	4.50000	−	7.79423i	0.244047	−	0.422701i
$341$	−27.7128			−1.50073
$342$		0			0
$343$	−20.0000			−1.07990
$344$	1.73205	−	3.00000i	0.0933859	−	0.161749i
$345$		0			0
$346$	16.5000	+	28.5788i	0.887045	+	1.53641i
$347$	−1.73205	−	3.00000i	−0.0929814	−	0.161048i	0.815783	−	0.578358i	$-0.196306\pi$
$347$	−1.73205	−	3.00000i	−0.0929814	−	0.161048i	−0.908764	+	0.417310i	$0.862973\pi$
$348$		0			0
$349$	−1.00000	+	1.73205i	−0.0535288	+	0.0927146i	−0.891548	−	0.452926i	$-0.850380\pi$
$349$	−1.00000	+	1.73205i	−0.0535288	+	0.0927146i	0.838019	+	0.545640i	$0.183714\pi$
$350$	−6.92820			−0.370328
$351$		0			0
$352$	18.0000			0.959403
$353$	6.92820	−	12.0000i	0.368751	−	0.638696i	−0.620620	−	0.784112i	$-0.713119\pi$
$353$	6.92820	−	12.0000i	0.368751	−	0.638696i	0.989371	+	0.145416i	$0.0464522\pi$
$354$		0			0
$355$	−9.00000	−	15.5885i	−0.477670	−	0.827349i
$356$	2.59808	+	4.50000i	0.137698	+	0.238500i
$357$		0			0
$358$	−18.0000	+	31.1769i	−0.951330	+	1.64775i
$359$	10.3923			0.548485			0.274242	−	0.961661i	$-0.411573\pi$
$359$	10.3923			0.548485			0.274242	+	0.961661i	$0.411573\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$	−1.73205	+	3.00000i	−0.0910346	+	0.157676i
$363$		0			0
$364$	1.00000	+	1.73205i	0.0524142	+	0.0907841i
$365$	−6.06218	−	10.5000i	−0.317309	−	0.549595i
$366$		0			0
$367$	−10.0000	+	17.3205i	−0.521996	+	0.904123i	0.477677	+	0.878536i	$0.341479\pi$
$367$	−10.0000	+	17.3205i	−0.521996	+	0.904123i	−0.999673	+	0.0255875i	$0.991854\pi$
$368$	−17.3205			−0.902894
$369$		0			0
$370$	21.0000			1.09174
$371$		0			0
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$	15.5885	+	27.0000i	0.806060	+	1.39614i
$375$		0			0
$376$	6.00000	−	10.3923i	0.309426	−	0.535942i
$377$	−1.73205			−0.0892052
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$	1.73205	−	3.00000i	0.0888523	−	0.153897i
$381$		0			0
$382$	−15.0000	−	25.9808i	−0.767467	−	1.32929i
$383$	8.66025	+	15.0000i	0.442518	+	0.766464i	0.997876	−	0.0651476i	$-0.0207518\pi$
$383$	8.66025	+	15.0000i	0.442518	+	0.766464i	−0.555357	+	0.831612i	$0.687419\pi$
$384$		0			0
$385$	−6.00000	+	10.3923i	−0.305788	+	0.529641i
$386$	−1.73205			−0.0881591
$387$		0			0
$388$	2.00000			0.101535
$389$	−13.8564	+	24.0000i	−0.702548	+	1.21685i	0.265022	+	0.964242i	$0.414621\pi$
$389$	−13.8564	+	24.0000i	−0.702548	+	1.21685i	−0.967569	+	0.252606i	$0.918712\pi$
$390$		0			0
$391$	−9.00000	−	15.5885i	−0.455150	−	0.788342i
$392$	−2.59808	−	4.50000i	−0.131223	−	0.227284i
$393$		0			0
$394$	4.50000	−	7.79423i	0.226707	−	0.392668i
$395$	−3.46410			−0.174298
$396$		0			0
$397$	29.0000			1.45547			0.727734	−	0.685859i	$-0.240573\pi$
$397$	29.0000			1.45547			0.727734	+	0.685859i	$0.240573\pi$
$398$	−17.3205	+	30.0000i	−0.868199	+	1.50376i
$399$		0			0
$400$	−5.00000	−	8.66025i	−0.250000	−	0.433013i
$401$	6.06218	+	10.5000i	0.302731	+	0.524345i	0.976753	−	0.214366i	$-0.0687684\pi$
$401$	6.06218	+	10.5000i	0.302731	+	0.524345i	−0.674023	+	0.738711i	$0.735435\pi$
$402$		0			0
$403$	4.00000	−	6.92820i	0.199254	−	0.345118i
$404$	6.92820			0.344691
$405$		0			0
$406$	6.00000			0.297775
$407$	−12.1244	+	21.0000i	−0.600982	+	1.04093i
$408$		0			0
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	0.999402	−	0.0345902i	$-0.0110126\pi$
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	−0.529657	+	0.848212i	$0.677679\pi$
$410$	−10.3923	−	18.0000i	−0.513239	−	0.888957i
$411$		0			0
$412$	−4.00000	+	6.92820i	−0.197066	+	0.341328i
$413$	27.7128			1.36366
$414$		0			0
$415$	24.0000			1.17811
$416$	−2.59808	+	4.50000i	−0.127381	+	0.220631i
$417$		0			0
$418$	6.00000	+	10.3923i	0.293470	+	0.508304i
$419$	3.46410	+	6.00000i	0.169232	+	0.293119i	0.938150	−	0.346228i	$-0.112538\pi$
$419$	3.46410	+	6.00000i	0.169232	+	0.293119i	−0.768918	+	0.639348i	$0.779204\pi$
$420$		0			0
$421$	12.5000	−	21.6506i	0.609213	−	1.05519i	−0.382158	−	0.924097i	$-0.624819\pi$
$421$	12.5000	−	21.6506i	0.609213	−	1.05519i	0.991370	−	0.131090i	$-0.0418478\pi$
$422$	−17.3205			−0.843149
$423$		0			0
$424$		0			0
$425$	5.19615	−	9.00000i	0.252050	−	0.436564i
$426$		0			0
$427$	7.00000	+	12.1244i	0.338754	+	0.586739i
$428$		0			0
$429$		0			0
$430$	3.00000	−	5.19615i	0.144673	−	0.250581i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	11.0000			0.528626			0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000			0.528626			0.264313	+	0.964437i	$0.414855\pi$
$434$	−13.8564	+	24.0000i	−0.665129	+	1.15204i
$435$		0			0
$436$	−5.50000	−	9.52628i	−0.263402	−	0.456226i
$437$	−3.46410	−	6.00000i	−0.165710	−	0.287019i
$438$		0			0
$439$	−10.0000	+	17.3205i	−0.477274	+	0.826663i	−0.999661	−	0.0260459i	$-0.991708\pi$
$439$	−10.0000	+	17.3205i	−0.477274	+	0.826663i	0.522387	+	0.852709i	$0.325042\pi$
$440$	−10.3923			−0.495434
$441$		0			0
$442$	−9.00000			−0.428086
$443$	17.3205	−	30.0000i	0.822922	−	1.42534i	−0.0805748	−	0.996749i	$-0.525676\pi$
$443$	17.3205	−	30.0000i	0.822922	−	1.42534i	0.903497	−	0.428594i	$-0.140991\pi$
$444$		0			0
$445$	−4.50000	−	7.79423i	−0.213320	−	0.369482i
$446$	−1.73205	−	3.00000i	−0.0820150	−	0.142054i
$447$		0			0
$448$	−1.00000	+	1.73205i	−0.0472456	+	0.0818317i
$449$	20.7846			0.980886			0.490443	−	0.871473i	$-0.336835\pi$
$449$	20.7846			0.980886			0.490443	+	0.871473i	$0.336835\pi$
$450$		0			0
$451$	24.0000			1.13012
$452$	0.866025	−	1.50000i	0.0407344	−	0.0705541i
$453$		0			0
$454$	3.00000	+	5.19615i	0.140797	+	0.243868i
$455$	−1.73205	−	3.00000i	−0.0811998	−	0.140642i
$456$		0			0
$457$	−14.5000	+	25.1147i	−0.678281	+	1.17482i	0.297217	+	0.954810i	$0.403942\pi$
$457$	−14.5000	+	25.1147i	−0.678281	+	1.17482i	−0.975498	+	0.220008i	$0.929392\pi$
$458$	−1.73205			−0.0809334
$459$		0			0
$460$	−6.00000			−0.279751
$461$	6.92820	−	12.0000i	0.322679	−	0.558896i	−0.658361	−	0.752702i	$-0.728750\pi$
$461$	6.92820	−	12.0000i	0.322679	−	0.558896i	0.981040	+	0.193806i	$0.0620834\pi$
$462$		0			0
$463$	−4.00000	−	6.92820i	−0.185896	−	0.321981i	0.757982	−	0.652275i	$-0.226185\pi$
$463$	−4.00000	−	6.92820i	−0.185896	−	0.321981i	−0.943878	+	0.330294i	$0.892852\pi$
$464$	4.33013	+	7.50000i	0.201021	+	0.348179i
$465$		0			0
$466$	22.5000	−	38.9711i	1.04229	−	1.80530i
$467$	−20.7846			−0.961797			−0.480899	−	0.876776i	$-0.659689\pi$
$467$	−20.7846			−0.961797			−0.480899	+	0.876776i	$0.659689\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$	10.3923	−	18.0000i	0.479361	−	0.830278i
$471$		0			0
$472$	12.0000	+	20.7846i	0.552345	+	0.956689i
$473$	3.46410	+	6.00000i	0.159280	+	0.275880i
$474$		0			0
$475$	2.00000	−	3.46410i	0.0917663	−	0.158944i
$476$	10.3923			0.476331
$477$		0			0
$478$	−48.0000			−2.19547
$479$	12.1244	−	21.0000i	0.553976	−	0.959514i	−0.444006	−	0.896024i	$-0.646443\pi$
$479$	12.1244	−	21.0000i	0.553976	−	0.959514i	0.997982	−	0.0634909i	$-0.0202234\pi$
$480$		0			0
$481$	−3.50000	−	6.06218i	−0.159586	−	0.276412i
$482$	−25.1147	−	43.5000i	−1.14394	−	1.98137i
$483$		0			0
$484$	−0.500000	+	0.866025i	−0.0227273	+	0.0393648i
$485$	−3.46410			−0.157297
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$	−6.06218	+	10.5000i	−0.274422	+	0.475313i
$489$		0			0
$490$	−4.50000	−	7.79423i	−0.203289	−	0.352107i
$491$	8.66025	+	15.0000i	0.390832	+	0.676941i	0.992559	−	0.121761i	$-0.0388541\pi$
$491$	8.66025	+	15.0000i	0.390832	+	0.676941i	−0.601728	+	0.798701i	$0.705521\pi$
$492$		0			0
$493$	−4.50000	+	7.79423i	−0.202670	+	0.351034i
$494$	−3.46410			−0.155857
$495$		0			0
$496$	−40.0000			−1.79605
$497$	10.3923	−	18.0000i	0.466159	−	0.807410i
$498$		0			0
$499$	5.00000	+	8.66025i	0.223831	+	0.387686i	0.955968	−	0.293471i	$-0.0948104\pi$
$499$	5.00000	+	8.66025i	0.223831	+	0.387686i	−0.732137	+	0.681157i	$0.761477\pi$
$500$	−6.06218	−	10.5000i	−0.271109	−	0.469574i
$501$		0			0
$502$	−9.00000	+	15.5885i	−0.401690	+	0.695747i
$503$	20.7846			0.926740			0.463370	−	0.886165i	$-0.346640\pi$
$503$	20.7846			0.926740			0.463370	+	0.886165i	$0.346640\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$	10.3923	−	18.0000i	0.461994	−	0.800198i
$507$		0			0
$508$	−1.00000	−	1.73205i	−0.0443678	−	0.0768473i
$509$	13.8564	+	24.0000i	0.614174	+	1.06378i	0.990529	+	0.137305i	$0.0438442\pi$
$509$	13.8564	+	24.0000i	0.614174	+	1.06378i	−0.376354	+	0.926476i	$0.622822\pi$
$510$		0			0
$511$	7.00000	−	12.1244i	0.309662	−	0.536350i
$512$	8.66025			0.382733
$513$		0			0
$514$	15.0000			0.661622
$515$	6.92820	−	12.0000i	0.305293	−	0.528783i
$516$		0			0
$517$	12.0000	+	20.7846i	0.527759	+	0.914106i
$518$	12.1244	+	21.0000i	0.532714	+	0.922687i
$519$		0			0
$520$	1.50000	−	2.59808i	0.0657794	−	0.113933i
$521$	−20.7846			−0.910590			−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846			−0.910590			−0.455295	+	0.890341i	$0.650466\pi$
$522$		0			0
$523$	38.0000			1.66162			0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000			1.66162			0.830812	+	0.556553i	$0.187876\pi$
$524$	−1.73205	+	3.00000i	−0.0756650	+	0.131056i
$525$		0			0
$526$	−6.00000	−	10.3923i	−0.261612	−	0.453126i
$527$	−20.7846	−	36.0000i	−0.905392	−	1.56818i
$528$		0			0
$529$	5.50000	−	9.52628i	0.239130	−	0.414186i
$530$		0			0
$531$		0			0
$532$	4.00000			0.173422
$533$	−3.46410	+	6.00000i	−0.150047	+	0.259889i
$534$		0			0
$535$		0			0
$536$	−8.66025	−	15.0000i	−0.374066	−	0.647901i
$537$		0			0
$538$	13.5000	−	23.3827i	0.582026	−	1.00810i
$539$	10.3923			0.447628
$540$		0			0
$541$	11.0000			0.472927			0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000			0.472927			0.236463	+	0.971640i	$0.424012\pi$
$542$	−1.73205	+	3.00000i	−0.0743980	+	0.128861i
$543$		0			0
$544$	13.5000	+	23.3827i	0.578808	+	1.00252i
$545$	9.52628	+	16.5000i	0.408061	+	0.706782i
$546$		0			0
$547$	−10.0000	+	17.3205i	−0.427569	+	0.740571i	−0.996657	−	0.0817056i	$-0.973963\pi$
$547$	−10.0000	+	17.3205i	−0.427569	+	0.740571i	0.569087	+	0.822277i	$0.307297\pi$
$548$	1.73205			0.0739895
$549$		0			0
$550$	12.0000			0.511682
$551$	−1.73205	+	3.00000i	−0.0737878	+	0.127804i
$552$		0			0
$553$	−2.00000	−	3.46410i	−0.0850487	−	0.147309i
$554$	−1.73205	−	3.00000i	−0.0735878	−	0.127458i
$555$		0			0
$556$	−4.00000	+	6.92820i	−0.169638	+	0.293821i
$557$	36.3731			1.54118			0.770588	−	0.637333i	$-0.219963\pi$
$557$	36.3731			1.54118			0.770588	+	0.637333i	$0.219963\pi$
$558$		0			0
$559$	−2.00000			−0.0845910
$560$	−8.66025	+	15.0000i	−0.365963	+	0.633866i
$561$		0			0
$562$	−10.5000	−	18.1865i	−0.442916	−	0.767153i
$563$	−17.3205	−	30.0000i	−0.729972	−	1.26435i	−0.956894	−	0.290436i	$-0.906200\pi$
$563$	−17.3205	−	30.0000i	−0.729972	−	1.26435i	0.226922	−	0.973913i	$-0.427134\pi$
$564$		0			0
$565$	−1.50000	+	2.59808i	−0.0631055	+	0.109302i
$566$	−48.4974			−2.03850
$567$		0			0
$568$	18.0000			0.755263
$569$	−16.4545	+	28.5000i	−0.689808	+	1.19478i	0.282092	+	0.959387i	$0.408972\pi$
$569$	−16.4545	+	28.5000i	−0.689808	+	1.19478i	−0.971900	+	0.235395i	$0.924362\pi$
$570$		0			0
$571$	−4.00000	−	6.92820i	−0.167395	−	0.289936i	0.770108	−	0.637913i	$-0.220202\pi$
$571$	−4.00000	−	6.92820i	−0.167395	−	0.289936i	−0.937503	+	0.347977i	$0.886869\pi$
$572$	−1.73205	−	3.00000i	−0.0724207	−	0.125436i
$573$		0			0
$574$	12.0000	−	20.7846i	0.500870	−	0.867533i
$575$	−6.92820			−0.288926
$576$		0			0
$577$	11.0000			0.457936			0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000			0.457936			0.228968	+	0.973434i	$0.426465\pi$
$578$	−8.66025	+	15.0000i	−0.360219	+	0.623918i
$579$		0			0
$580$	1.50000	+	2.59808i	0.0622841	+	0.107879i
$581$	13.8564	+	24.0000i	0.574861	+	0.995688i
$582$		0			0
$583$		0			0
$584$	12.1244			0.501709
$585$		0			0
$586$	33.0000			1.36322
$587$	−19.0526	+	33.0000i	−0.786383	+	1.36206i	0.141786	+	0.989897i	$0.454716\pi$
$587$	−19.0526	+	33.0000i	−0.786383	+	1.36206i	−0.928169	+	0.372158i	$0.878618\pi$
$588$		0			0
$589$	−8.00000	−	13.8564i	−0.329634	−	0.570943i
$590$	20.7846	+	36.0000i	0.855689	+	1.48210i
$591$		0			0
$592$	−17.5000	+	30.3109i	−0.719246	+	1.24577i
$593$	15.5885			0.640141			0.320071	−	0.947394i	$-0.396293\pi$
$593$	15.5885			0.640141			0.320071	+	0.947394i	$0.396293\pi$
$594$		0			0
$595$	−18.0000			−0.737928
$596$	−4.33013	+	7.50000i	−0.177369	+	0.307212i
$597$		0			0
$598$	3.00000	+	5.19615i	0.122679	+	0.212486i
$599$	−6.92820	−	12.0000i	−0.283079	−	0.490307i	0.689063	−	0.724702i	$-0.258022\pi$
$599$	−6.92820	−	12.0000i	−0.283079	−	0.490307i	−0.972141	+	0.234395i	$0.924689\pi$
$600$		0			0
$601$	12.5000	−	21.6506i	0.509886	−	0.883148i	−0.490049	−	0.871695i	$-0.663021\pi$
$601$	12.5000	−	21.6506i	0.509886	−	0.883148i	0.999934	−	0.0114528i	$-0.00364562\pi$
$602$	6.92820			0.282372
$603$		0			0
$604$	20.0000			0.813788
$605$	0.866025	−	1.50000i	0.0352089	−	0.0609837i
$606$		0			0
$607$	−13.0000	−	22.5167i	−0.527654	−	0.913923i	−0.999480	−	0.0322317i	$-0.989739\pi$
$607$	−13.0000	−	22.5167i	−0.527654	−	0.913923i	0.471827	−	0.881691i	$-0.343595\pi$
$608$	5.19615	+	9.00000i	0.210732	+	0.364998i
$609$		0			0
$610$	−10.5000	+	18.1865i	−0.425133	+	0.736351i
$611$	−6.92820			−0.280285
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$	13.8564	−	24.0000i	0.559199	−	0.968561i
$615$		0			0
$616$	−6.00000	−	10.3923i	−0.241747	−	0.418718i
$617$	6.06218	+	10.5000i	0.244054	+	0.422714i	0.961865	−	0.273524i	$-0.0881892\pi$
$617$	6.06218	+	10.5000i	0.244054	+	0.422714i	−0.717811	+	0.696238i	$0.754856\pi$
$618$		0			0
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	−0.993959	−	0.109749i	$-0.964995\pi$
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	0.592025	+	0.805919i	$0.298329\pi$
$620$	−13.8564			−0.556487
$621$		0			0
$622$	−12.0000			−0.481156
$623$	5.19615	−	9.00000i	0.208179	−	0.360577i
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$	21.6506	+	37.5000i	0.865333	+	1.49880i
$627$		0			0
$628$	−8.50000	+	14.7224i	−0.339187	+	0.587489i
$629$	−36.3731			−1.45029
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$	1.73205	−	3.00000i	0.0688973	−	0.119334i
$633$		0			0
$634$	7.50000	+	12.9904i	0.297863	+	0.515914i
$635$	1.73205	+	3.00000i	0.0687343	+	0.119051i
$636$		0			0
$637$	−1.50000	+	2.59808i	−0.0594322	+	0.102940i
$638$	−10.3923			−0.411435
$639$		0			0
$640$	−21.0000			−0.830098
$641$	−11.2583	+	19.5000i	−0.444677	+	0.770204i	−0.998030	−	0.0627436i	$-0.980015\pi$
$641$	−11.2583	+	19.5000i	−0.444677	+	0.770204i	0.553352	+	0.832947i	$0.313348\pi$
$642$		0			0
$643$	−4.00000	−	6.92820i	−0.157745	−	0.273222i	0.776310	−	0.630351i	$-0.217089\pi$
$643$	−4.00000	−	6.92820i	−0.157745	−	0.273222i	−0.934055	+	0.357129i	$0.883756\pi$
$644$	−3.46410	−	6.00000i	−0.136505	−	0.236433i
$645$		0			0
$646$	−9.00000	+	15.5885i	−0.354100	+	0.613320i
$647$	−31.1769			−1.22569			−0.612845	−	0.790203i	$-0.709975\pi$
$647$	−31.1769			−1.22569			−0.612845	+	0.790203i	$0.709975\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$	−1.73205	+	3.00000i	−0.0679366	+	0.117670i
$651$		0			0
$652$	8.00000	+	13.8564i	0.313304	+	0.542659i
$653$	−6.92820	−	12.0000i	−0.271122	−	0.469596i	0.698028	−	0.716071i	$-0.254061\pi$
$653$	−6.92820	−	12.0000i	−0.271122	−	0.469596i	−0.969149	+	0.246474i	$0.920728\pi$
$654$		0			0
$655$	3.00000	−	5.19615i	0.117220	−	0.203030i
$656$	34.6410			1.35250
$657$		0			0
$658$	24.0000			0.935617
$659$	1.73205	−	3.00000i	0.0674711	−	0.116863i	−0.830316	−	0.557292i	$-0.811840\pi$
$659$	1.73205	−	3.00000i	0.0674711	−	0.116863i	0.897787	+	0.440429i	$0.145174\pi$
$660$		0			0
$661$	−8.50000	−	14.7224i	−0.330612	−	0.572636i	0.652020	−	0.758202i	$-0.273922\pi$
$661$	−8.50000	−	14.7224i	−0.330612	−	0.572636i	−0.982632	+	0.185565i	$0.940588\pi$
$662$	−1.73205	−	3.00000i	−0.0673181	−	0.116598i
$663$		0			0
$664$	−12.0000	+	20.7846i	−0.465690	+	0.806599i
$665$	−6.92820			−0.268664
$666$		0			0
$667$	6.00000			0.232321
$668$	8.66025	−	15.0000i	0.335075	−	0.580367i
$669$		0			0
$670$	−15.0000	−	25.9808i	−0.579501	−	1.00372i
$671$	−12.1244	−	21.0000i	−0.468056	−	0.810696i
$672$		0			0
$673$	12.5000	−	21.6506i	0.481840	−	0.834571i	−0.517943	−	0.855415i	$-0.673302\pi$
$673$	12.5000	−	21.6506i	0.481840	−	0.834571i	0.999783	+	0.0208444i	$0.00663546\pi$
$674$	45.0333			1.73462
$675$		0			0
$676$	−12.0000			−0.461538
$677$	6.92820	−	12.0000i	0.266272	−	0.461197i	−0.701624	−	0.712548i	$-0.747541\pi$
$677$	6.92820	−	12.0000i	0.266272	−	0.461197i	0.967896	+	0.251350i	$0.0808746\pi$
$678$		0			0
$679$	−2.00000	−	3.46410i	−0.0767530	−	0.132940i
$680$	−7.79423	−	13.5000i	−0.298895	−	0.517701i
$681$		0			0
$682$	24.0000	−	41.5692i	0.919007	−	1.59177i
$683$	−20.7846			−0.795301			−0.397650	−	0.917537i	$-0.630174\pi$
$683$	−20.7846			−0.795301			−0.397650	+	0.917537i	$0.630174\pi$
$684$		0			0
$685$	−3.00000			−0.114624
$686$	17.3205	−	30.0000i	0.661300	−	1.14541i
$687$		0			0
$688$	5.00000	+	8.66025i	0.190623	+	0.330169i
$689$		0			0
$690$		0			0
$691$	−1.00000	+	1.73205i	−0.0380418	+	0.0658903i	−0.884419	−	0.466693i	$-0.845445\pi$
$691$	−1.00000	+	1.73205i	−0.0380418	+	0.0658903i	0.846378	+	0.532583i	$0.178779\pi$
$692$	−19.0526			−0.724270
$693$		0			0
$694$	6.00000			0.227757
$695$	6.92820	−	12.0000i	0.262802	−	0.455186i
$696$		0			0
$697$	18.0000	+	31.1769i	0.681799	+	1.18091i
$698$	−1.73205	−	3.00000i	−0.0655591	−	0.113552i
$699$		0			0
$700$	2.00000	−	3.46410i	0.0755929	−	0.130931i
$701$	46.7654			1.76630			0.883152	−	0.469087i	$-0.155417\pi$
$701$	46.7654			1.76630			0.883152	+	0.469087i	$0.155417\pi$
$702$		0			0
$703$	−14.0000			−0.528020
$704$	1.73205	−	3.00000i	0.0652791	−	0.113067i
$705$		0			0
$706$	12.0000	+	20.7846i	0.451626	+	0.782239i
$707$	−6.92820	−	12.0000i	−0.260562	−	0.451306i
$708$		0			0
$709$	12.5000	−	21.6506i	0.469447	−	0.813107i	−0.529943	−	0.848034i	$-0.677787\pi$
$709$	12.5000	−	21.6506i	0.469447	−	0.813107i	0.999390	+	0.0349269i	$0.0111198\pi$
$710$	31.1769			1.17005
$711$		0			0
$712$	9.00000			0.337289
$713$	−13.8564	+	24.0000i	−0.518927	+	0.898807i
$714$		0			0
$715$	3.00000	+	5.19615i	0.112194	+	0.194325i
$716$	−10.3923	−	18.0000i	−0.388379	−	0.672692i
$717$		0			0
$718$	−9.00000	+	15.5885i	−0.335877	+	0.581756i
$719$	−10.3923			−0.387568			−0.193784	−	0.981044i	$-0.562076\pi$
$719$	−10.3923			−0.387568			−0.193784	+	0.981044i	$0.562076\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$	12.9904	−	22.5000i	0.483452	−	0.837363i
$723$		0			0
$724$	−1.00000	−	1.73205i	−0.0371647	−	0.0643712i
$725$	1.73205	+	3.00000i	0.0643268	+	0.111417i
$726$		0			0
$727$	17.0000	−	29.4449i	0.630495	−	1.09205i	−0.356956	−	0.934121i	$-0.616185\pi$
$727$	17.0000	−	29.4449i	0.630495	−	1.09205i	0.987451	−	0.157928i	$-0.0504814\pi$
$728$	3.46410			0.128388
$729$		0			0
$730$	21.0000			0.777245
$731$	−5.19615	+	9.00000i	−0.192187	+	0.332877i
$732$		0			0
$733$	23.0000	+	39.8372i	0.849524	+	1.47142i	0.881633	+	0.471935i	$0.156444\pi$
$733$	23.0000	+	39.8372i	0.849524	+	1.47142i	−0.0321090	+	0.999484i	$0.510222\pi$
$734$	−17.3205	−	30.0000i	−0.639312	−	1.10732i
$735$		0			0
$736$	9.00000	−	15.5885i	0.331744	−	0.574598i
$737$	34.6410			1.27602
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	−6.06218	+	10.5000i	−0.222850	+	0.385988i
$741$		0			0
$742$		0			0
$743$	3.46410	+	6.00000i	0.127086	+	0.220119i	0.922546	−	0.385887i	$-0.126104\pi$
$743$	3.46410	+	6.00000i	0.127086	+	0.220119i	−0.795461	+	0.606005i	$0.792771\pi$
$744$		0			0
$745$	7.50000	−	12.9904i	0.274779	−	0.475931i
$746$	−17.3205			−0.634149
$747$		0			0
$748$	−18.0000			−0.658145
$749$		0			0
$750$		0			0
$751$	5.00000	+	8.66025i	0.182453	+	0.316017i	0.942715	−	0.333599i	$-0.108263\pi$
$751$	5.00000	+	8.66025i	0.182453	+	0.316017i	−0.760263	+	0.649616i	$0.774930\pi$
$752$	17.3205	+	30.0000i	0.631614	+	1.09399i
$753$		0			0
$754$	1.50000	−	2.59808i	0.0546268	−	0.0946164i
$755$	−34.6410			−1.26072
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	13.8564	−	24.0000i	0.503287	−	0.871719i
$759$		0			0
$760$	−3.00000	−	5.19615i	−0.108821	−	0.188484i
$761$	−14.7224	−	25.5000i	−0.533688	−	0.924374i	−0.999226	−	0.0393463i	$-0.987472\pi$
$761$	−14.7224	−	25.5000i	−0.533688	−	0.924374i	0.465538	−	0.885028i	$-0.345861\pi$
$762$		0			0
$763$	−11.0000	+	19.0526i	−0.398227	+	0.689749i
$764$	17.3205			0.626634
$765$		0			0
$766$	−30.0000			−1.08394
$767$	6.92820	−	12.0000i	0.250163	−	0.433295i
$768$		0			0
$769$	0.500000	+	0.866025i	0.0180305	+	0.0312297i	0.874900	−	0.484304i	$-0.160927\pi$
$769$	0.500000	+	0.866025i	0.0180305	+	0.0312297i	−0.856869	+	0.515534i	$0.827594\pi$
$770$	−10.3923	−	18.0000i	−0.374513	−	0.648675i
$771$		0			0
$772$	0.500000	−	0.866025i	0.0179954	−	0.0311689i
$773$	−25.9808			−0.934463			−0.467232	−	0.884135i	$-0.654749\pi$
$773$	−25.9808			−0.934463			−0.467232	+	0.884135i	$0.654749\pi$
$774$		0			0
$775$	−16.0000			−0.574737
$776$	1.73205	−	3.00000i	0.0621770	−	0.107694i
$777$		0			0
$778$	−24.0000	−	41.5692i	−0.860442	−	1.49033i
$779$	6.92820	+	12.0000i	0.248229	+	0.429945i
$780$		0			0
$781$	−18.0000	+	31.1769i	−0.644091	+	1.11560i
$782$	31.1769			1.11488
$783$		0			0
$784$	15.0000			0.535714
$785$	14.7224	−	25.5000i	0.525466	−	0.910134i
$786$		0			0
$787$	−13.0000	−	22.5167i	−0.463400	−	0.802632i	0.535728	−	0.844391i	$-0.320037\pi$
$787$	−13.0000	−	22.5167i	−0.463400	−	0.802632i	−0.999128	+	0.0417585i	$0.986704\pi$
$788$	2.59808	+	4.50000i	0.0925526	+	0.160306i
$789$		0			0
$790$	3.00000	−	5.19615i	0.106735	−	0.184871i
$791$	−3.46410			−0.123169
$792$		0			0
$793$	7.00000			0.248577
$794$	−25.1147	+	43.5000i	−0.891289	+	1.54376i
$795$		0			0
$796$	−10.0000	−	17.3205i	−0.354441	−	0.613909i
$797$	26.8468	+	46.5000i	0.950962	+	1.64711i	0.743349	+	0.668903i	$0.233236\pi$
$797$	26.8468	+	46.5000i	0.950962	+	1.64711i	0.207613	+	0.978211i	$0.433431\pi$
$798$		0			0
$799$	−18.0000	+	31.1769i	−0.636794	+	1.10296i
$800$	10.3923			0.367423
$801$		0			0
$802$	−21.0000			−0.741536
$803$	−12.1244	+	21.0000i	−0.427859	+	0.741074i
$804$		0			0
$805$	6.00000	+	10.3923i	0.211472	+	0.366281i
$806$	6.92820	+	12.0000i	0.244036	+	0.422682i
$807$		0			0
$808$	6.00000	−	10.3923i	0.211079	−	0.365600i
$809$	−46.7654			−1.64418			−0.822091	−	0.569355i	$-0.807193\pi$
$809$	−46.7654			−1.64418			−0.822091	+	0.569355i	$0.807193\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$	−1.73205	+	3.00000i	−0.0607831	+	0.105279i
$813$		0			0
$814$	−21.0000	−	36.3731i	−0.736050	−	1.27488i
$815$	−13.8564	−	24.0000i	−0.485369	−	0.840683i
$816$		0			0
$817$	−2.00000	+	3.46410i	−0.0699711	+	0.121194i
$818$	−32.9090			−1.15063
$819$		0			0
$820$	12.0000			0.419058
$821$	−6.06218	+	10.5000i	−0.211571	+	0.366453i	−0.952207	−	0.305455i	$-0.901191\pi$
$821$	−6.06218	+	10.5000i	−0.211571	+	0.366453i	0.740635	+	0.671908i	$0.234525\pi$
$822$		0			0
$823$	14.0000	+	24.2487i	0.488009	+	0.845257i	0.999905	−	0.0137907i	$-0.00438987\pi$
$823$	14.0000	+	24.2487i	0.488009	+	0.845257i	−0.511896	+	0.859048i	$0.671057\pi$
$824$	6.92820	+	12.0000i	0.241355	+	0.418040i
$825$		0			0
$826$	−24.0000	+	41.5692i	−0.835067	+	1.44638i
$827$	−10.3923			−0.361376			−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923			−0.361376			−0.180688	+	0.983540i	$0.557832\pi$
$828$		0			0
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$	−20.7846	+	36.0000i	−0.721444	+	1.24958i
$831$		0			0
$832$	0.500000	+	0.866025i	0.0173344	+	0.0300240i
$833$	7.79423	+	13.5000i	0.270054	+	0.467747i
$834$		0			0
$835$	−15.0000	+	25.9808i	−0.519096	+	0.899101i
$836$	−6.92820			−0.239617
$837$		0			0
$838$	−12.0000			−0.414533
$839$	22.5167	−	39.0000i	0.777361	−	1.34643i	−0.156096	−	0.987742i	$-0.549891\pi$
$839$	22.5167	−	39.0000i	0.777361	−	1.34643i	0.933458	−	0.358688i	$-0.116776\pi$
$840$		0			0
$841$	13.0000	+	22.5167i	0.448276	+	0.776437i
$842$	21.6506	+	37.5000i	0.746130	+	1.29234i
$843$		0			0
$844$	5.00000	−	8.66025i	0.172107	−	0.298098i
$845$	20.7846			0.715012
$846$		0			0
$847$	2.00000			0.0687208
$848$		0			0
$849$		0			0
$850$	9.00000	+	15.5885i	0.308697	+	0.534680i
$851$	12.1244	+	21.0000i	0.415618	+	0.719871i
$852$		0			0
$853$	17.0000	−	29.4449i	0.582069	−	1.00817i	−0.413165	−	0.910656i	$-0.635577\pi$
$853$	17.0000	−	29.4449i	0.582069	−	1.00817i	0.995234	−	0.0975167i	$-0.0310899\pi$
$854$	−24.2487			−0.829774
$855$		0			0
$856$		0			0
$857$	−11.2583	+	19.5000i	−0.384577	+	0.666107i	−0.991710	−	0.128493i	$-0.958986\pi$
$857$	−11.2583	+	19.5000i	−0.384577	+	0.666107i	0.607133	+	0.794600i	$0.292319\pi$
$858$		0			0
$859$	−22.0000	−	38.1051i	−0.750630	−	1.30013i	−0.947518	−	0.319704i	$-0.896417\pi$
$859$	−22.0000	−	38.1051i	−0.750630	−	1.30013i	0.196887	−	0.980426i	$-0.436917\pi$
$860$	1.73205	+	3.00000i	0.0590624	+	0.102299i
$861$		0			0
$862$		0			0
$863$	31.1769			1.06127			0.530637	−	0.847599i	$-0.321953\pi$
$863$	31.1769			1.06127			0.530637	+	0.847599i	$0.321953\pi$
$864$		0			0
$865$	33.0000			1.12203
$866$	−9.52628	+	16.5000i	−0.323716	+	0.560693i
$867$		0			0
$868$	−8.00000	−	13.8564i	−0.271538	−	0.470317i
$869$	3.46410	+	6.00000i	0.117512	+	0.203536i
$870$		0			0
$871$	−5.00000	+	8.66025i	−0.169419	+	0.293442i
$872$	−19.0526			−0.645201
$873$		0			0
$874$	12.0000			0.405906
$875$	−12.1244	+	21.0000i	−0.409878	+	0.709930i
$876$		0			0
$877$	−26.5000	−	45.8993i	−0.894841	−	1.54991i	−0.834001	−	0.551763i	$-0.813955\pi$
$877$	−26.5000	−	45.8993i	−0.894841	−	1.54991i	−0.0608407	−	0.998147i	$-0.519378\pi$
$878$	−17.3205	−	30.0000i	−0.584539	−	1.01245i
$879$		0			0
$880$	15.0000	−	25.9808i	0.505650	−	0.875811i
$881$	20.7846			0.700251			0.350126	−	0.936703i	$-0.386139\pi$
$881$	20.7846			0.700251			0.350126	+	0.936703i	$0.386139\pi$
$882$		0			0
$883$	56.0000			1.88455			0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000			1.88455			0.942275	+	0.334840i	$0.108682\pi$
$884$	2.59808	−	4.50000i	0.0873828	−	0.151351i
$885$		0			0
$886$	30.0000	+	51.9615i	1.00787	+	1.74568i
$887$	−1.73205	−	3.00000i	−0.0581566	−	0.100730i	0.835481	−	0.549519i	$-0.185189\pi$
$887$	−1.73205	−	3.00000i	−0.0581566	−	0.100730i	−0.893638	+	0.448789i	$0.851856\pi$
$888$		0			0
$889$	−2.00000	+	3.46410i	−0.0670778	+	0.116182i
$890$	15.5885			0.522526
$891$		0			0
$892$	2.00000			0.0669650
$893$	−6.92820	+	12.0000i	−0.231843	+	0.401565i
$894$		0			0
$895$	18.0000	+	31.1769i	0.601674	+	1.04213i
$896$	−12.1244	−	21.0000i	−0.405046	−	0.701561i
$897$		0			0
$898$	−18.0000	+	31.1769i	−0.600668	+	1.04039i
$899$	13.8564			0.462137
$900$		0			0
$901$		0			0
$902$	−20.7846	+	36.0000i	−0.692052	+	1.19867i
$903$		0			0
$904$	−1.50000	−	2.59808i	−0.0498893	−	0.0864107i
$905$	1.73205	+	3.00000i	0.0575753	+	0.0997234i
$906$		0			0
$907$	26.0000	−	45.0333i	0.863316	−	1.49531i	−0.00539395	−	0.999985i	$-0.501717\pi$
$907$	26.0000	−	45.0333i	0.863316	−	1.49531i	0.868710	−	0.495321i	$-0.164950\pi$
$908$	−3.46410			−0.114960
$909$		0			0
$910$	6.00000			0.198898
$911$	12.1244	−	21.0000i	0.401698	−	0.695761i	−0.592233	−	0.805767i	$-0.701754\pi$
$911$	12.1244	−	21.0000i	0.401698	−	0.695761i	0.993931	+	0.110006i	$0.0350869\pi$
$912$		0			0
$913$	−24.0000	−	41.5692i	−0.794284	−	1.37574i
$914$	−25.1147	−	43.5000i	−0.830722	−	1.43885i
$915$		0			0
$916$	0.500000	−	0.866025i	0.0165205	−	0.0286143i
$917$	6.92820			0.228789
$918$		0			0
$919$	2.00000			0.0659739			0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000			0.0659739			0.0329870	+	0.999456i	$0.489498\pi$
$920$	−5.19615	+	9.00000i	−0.171312	+	0.296721i
$921$		0			0
$922$	12.0000	+	20.7846i	0.395199	+	0.684505i
$923$	−5.19615	−	9.00000i	−0.171033	−	0.296239i
$924$		0			0
$925$	−7.00000	+	12.1244i	−0.230159	+	0.398646i
$926$	13.8564			0.455350
$927$		0			0
$928$	−9.00000			−0.295439
$929$	25.1147	−	43.5000i	0.823988	−	1.42719i	−0.0787027	−	0.996898i	$-0.525078\pi$
$929$	25.1147	−	43.5000i	0.823988	−	1.42719i	0.902690	−	0.430291i	$-0.141589\pi$
$930$		0			0
$931$	3.00000	+	5.19615i	0.0983210	+	0.170297i
$932$	12.9904	+	22.5000i	0.425514	+	0.737012i
$933$		0			0
$934$	18.0000	−	31.1769i	0.588978	−	1.02014i
$935$	31.1769			1.01959
$936$		0			0
$937$	−25.0000			−0.816714			−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000			−0.816714			−0.408357	+	0.912822i	$0.633898\pi$
$938$	17.3205	−	30.0000i	0.565535	−	0.979535i
$939$		0			0
$940$	6.00000	+	10.3923i	0.195698	+	0.338960i
$941$	−25.1147	−	43.5000i	−0.818717	−	1.41806i	−0.906628	−	0.421931i	$-0.861352\pi$
$941$	−25.1147	−	43.5000i	−0.818717	−	1.41806i	0.0879109	−	0.996128i	$-0.471981\pi$
$942$		0			0
$943$	12.0000	−	20.7846i	0.390774	−	0.676840i
$944$	−69.2820			−2.25494
$945$		0			0
$946$	−12.0000			−0.390154
$947$	−8.66025	+	15.0000i	−0.281420	+	0.487435i	−0.971735	−	0.236075i	$-0.924139\pi$
$947$	−8.66025	+	15.0000i	−0.281420	+	0.487435i	0.690314	+	0.723510i	$0.257472\pi$
$948$		0			0
$949$	−3.50000	−	6.06218i	−0.113615	−	0.196787i
$950$	3.46410	+	6.00000i	0.112390	+	0.194666i
$951$		0			0
$952$	9.00000	−	15.5885i	0.291692	−	0.505225i
$953$	−5.19615			−0.168320			−0.0841599	−	0.996452i	$-0.526821\pi$
$953$	−5.19615			−0.168320			−0.0841599	+	0.996452i	$0.526821\pi$
$954$		0			0
$955$	−30.0000			−0.970777
$956$	13.8564	−	24.0000i	0.448148	−	0.776215i
$957$		0			0
$958$	21.0000	+	36.3731i	0.678479	+	1.17516i
$959$	−1.73205	−	3.00000i	−0.0559308	−	0.0968751i
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$	12.1244			0.390905
$963$		0			0
$964$	29.0000			0.934027
$965$	−0.866025	+	1.50000i	−0.0278783	+	0.0482867i
$966$		0			0
$967$	23.0000	+	39.8372i	0.739630	+	1.28108i	0.952662	+	0.304032i	$0.0983329\pi$
$967$	23.0000	+	39.8372i	0.739630	+	1.28108i	−0.213032	+	0.977045i	$0.568334\pi$
$968$	0.866025	+	1.50000i	0.0278351	+	0.0482118i
$969$		0			0
$970$	3.00000	−	5.19615i	0.0963242	−	0.166838i
$971$	31.1769			1.00051			0.500257	−	0.865877i	$-0.333239\pi$
$971$	31.1769			1.00051			0.500257	+	0.865877i	$0.333239\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$	13.8564	−	24.0000i	0.443988	−	0.769010i
$975$		0			0
$976$	−17.5000	−	30.3109i	−0.560161	−	0.970228i
$977$	24.2487	+	42.0000i	0.775785	+	1.34370i	0.934352	+	0.356351i	$0.115979\pi$
$977$	24.2487	+	42.0000i	0.775785	+	1.34370i	−0.158567	+	0.987348i	$0.550687\pi$
$978$		0			0
$979$	−9.00000	+	15.5885i	−0.287641	+	0.498209i
$980$	5.19615			0.165985
$981$		0			0
$982$	−30.0000			−0.957338
$983$	17.3205	−	30.0000i	0.552438	−	0.956851i	−0.445659	−	0.895203i	$-0.647031\pi$
$983$	17.3205	−	30.0000i	0.552438	−	0.956851i	0.998098	−	0.0616488i	$-0.0196359\pi$
$984$		0			0
$985$	−4.50000	−	7.79423i	−0.143382	−	0.248345i
$986$	−7.79423	−	13.5000i	−0.248219	−	0.429928i
$987$		0			0
$988$	1.00000	−	1.73205i	0.0318142	−	0.0551039i
$989$	6.92820			0.220304
$990$		0			0
$991$	−34.0000			−1.08005			−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000			−1.08005			−0.540023	+	0.841650i	$0.681584\pi$
$992$	20.7846	−	36.0000i	0.659912	−	1.14300i
$993$		0			0
$994$	18.0000	+	31.1769i	0.570925	+	0.988872i
$995$	17.3205	+	30.0000i	0.549097	+	0.951064i
$996$		0			0
$997$	3.50000	−	6.06218i	0.110846	−	0.191991i	−0.805266	−	0.592914i	$-0.797977\pi$
$997$	3.50000	−	6.06218i	0.110846	−	0.191991i	0.916112	+	0.400923i	$0.131311\pi$
$998$	−17.3205			−0.548271
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.2.c.b.55.1		4
3.2	odd	2	inner	81.2.c.b.55.2		4
4.3	odd	2		1296.2.i.s.865.2		4
9.2	odd	6		81.2.a.a.1.1	✓	2
9.4	even	3	inner	81.2.c.b.28.1		4
9.5	odd	6	inner	81.2.c.b.28.2		4
9.7	even	3		81.2.a.a.1.2	yes	2
12.11	even	2		1296.2.i.s.865.1		4
27.2	odd	18		729.2.e.o.325.1		12
27.4	even	9		729.2.e.o.163.1		12
27.5	odd	18		729.2.e.o.406.1		12
27.7	even	9		729.2.e.o.82.2		12
27.11	odd	18		729.2.e.o.568.2		12
27.13	even	9		729.2.e.o.649.2		12
27.14	odd	18		729.2.e.o.649.1		12
27.16	even	9		729.2.e.o.568.1		12
27.20	odd	18		729.2.e.o.82.1		12
27.22	even	9		729.2.e.o.406.2		12
27.23	odd	18		729.2.e.o.163.2		12
27.25	even	9		729.2.e.o.325.2		12
36.7	odd	6		1296.2.a.o.1.1		2
36.11	even	6		1296.2.a.o.1.2		2
36.23	even	6		1296.2.i.s.433.1		4
36.31	odd	6		1296.2.i.s.433.2		4
45.2	even	12		2025.2.b.k.649.2		4
45.7	odd	12		2025.2.b.k.649.4		4
45.29	odd	6		2025.2.a.j.1.2		2
45.34	even	6		2025.2.a.j.1.1		2
45.38	even	12		2025.2.b.k.649.3		4
45.43	odd	12		2025.2.b.k.649.1		4
63.20	even	6		3969.2.a.i.1.1		2
63.34	odd	6		3969.2.a.i.1.2		2
72.11	even	6		5184.2.a.bq.1.1		2
72.29	odd	6		5184.2.a.br.1.1		2
72.43	odd	6		5184.2.a.bq.1.2		2
72.61	even	6		5184.2.a.br.1.2		2
99.43	odd	6		9801.2.a.v.1.1		2
99.65	even	6		9801.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.a.a.1.1	✓	2	9.2	odd	6
81.2.a.a.1.2	yes	2	9.7	even	3
81.2.c.b.28.1		4	9.4	even	3	inner
81.2.c.b.28.2		4	9.5	odd	6	inner
81.2.c.b.55.1		4	1.1	even	1	trivial
81.2.c.b.55.2		4	3.2	odd	2	inner
729.2.e.o.82.1		12	27.20	odd	18
729.2.e.o.82.2		12	27.7	even	9
729.2.e.o.163.1		12	27.4	even	9
729.2.e.o.163.2		12	27.23	odd	18
729.2.e.o.325.1		12	27.2	odd	18
729.2.e.o.325.2		12	27.25	even	9
729.2.e.o.406.1		12	27.5	odd	18
729.2.e.o.406.2		12	27.22	even	9
729.2.e.o.568.1		12	27.16	even	9
729.2.e.o.568.2		12	27.11	odd	18
729.2.e.o.649.1		12	27.14	odd	18
729.2.e.o.649.2		12	27.13	even	9
1296.2.a.o.1.1		2	36.7	odd	6
1296.2.a.o.1.2		2	36.11	even	6
1296.2.i.s.433.1		4	36.23	even	6
1296.2.i.s.433.2		4	36.31	odd	6
1296.2.i.s.865.1		4	12.11	even	2
1296.2.i.s.865.2		4	4.3	odd	2
2025.2.a.j.1.1		2	45.34	even	6
2025.2.a.j.1.2		2	45.29	odd	6
2025.2.b.k.649.1		4	45.43	odd	12
2025.2.b.k.649.2		4	45.2	even	12
2025.2.b.k.649.3		4	45.38	even	12
2025.2.b.k.649.4		4	45.7	odd	12
3969.2.a.i.1.1		2	63.20	even	6
3969.2.a.i.1.2		2	63.34	odd	6
5184.2.a.bq.1.1		2	72.11	even	6
5184.2.a.bq.1.2		2	72.43	odd	6
5184.2.a.br.1.1		2	72.29	odd	6
5184.2.a.br.1.2		2	72.61	even	6
9801.2.a.v.1.1		2	99.43	odd	6
9801.2.a.v.1.2		2	99.65	even	6

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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 1.50000i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.866025 + 1.50000i) q^{5} +(-1.00000 + 1.73205i) q^{7} -1.73205 q^{8} +O(q^{10})\)	\(q+(-0.866025 + 1.50000i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.866025 + 1.50000i) q^{5} +(-1.00000 + 1.73205i) q^{7} -1.73205 q^{8} -3.00000 q^{10} +(1.73205 - 3.00000i) q^{11} +(0.500000 + 0.866025i) q^{13} +(-1.73205 - 3.00000i) q^{14} +(2.50000 - 4.33013i) q^{16} +5.19615 q^{17} +2.00000 q^{19} +(0.866025 - 1.50000i) q^{20} +(3.00000 + 5.19615i) q^{22} +(-1.73205 - 3.00000i) q^{23} +(1.00000 - 1.73205i) q^{25} -1.73205 q^{26} +2.00000 q^{28} +(-0.866025 + 1.50000i) q^{29} +(-4.00000 - 6.92820i) q^{31} +(2.59808 + 4.50000i) q^{32} +(-4.50000 + 7.79423i) q^{34} -3.46410 q^{35} -7.00000 q^{37} +(-1.73205 + 3.00000i) q^{38} +(-1.50000 - 2.59808i) q^{40} +(3.46410 + 6.00000i) q^{41} +(-1.00000 + 1.73205i) q^{43} -3.46410 q^{44} +6.00000 q^{46} +(-3.46410 + 6.00000i) q^{47} +(1.50000 + 2.59808i) q^{49} +(1.73205 + 3.00000i) q^{50} +(0.500000 - 0.866025i) q^{52} +6.00000 q^{55} +(1.73205 - 3.00000i) q^{56} +(-1.50000 - 2.59808i) q^{58} +(-6.92820 - 12.0000i) q^{59} +(3.50000 - 6.06218i) q^{61} +13.8564 q^{62} +1.00000 q^{64} +(-0.866025 + 1.50000i) q^{65} +(5.00000 + 8.66025i) q^{67} +(-2.59808 - 4.50000i) q^{68} +(3.00000 - 5.19615i) q^{70} -10.3923 q^{71} -7.00000 q^{73} +(6.06218 - 10.5000i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(3.46410 + 6.00000i) q^{77} +(-1.00000 + 1.73205i) q^{79} +8.66025 q^{80} -12.0000 q^{82} +(6.92820 - 12.0000i) q^{83} +(4.50000 + 7.79423i) q^{85} +(-1.73205 - 3.00000i) q^{86} +(-3.00000 + 5.19615i) q^{88} -5.19615 q^{89} -2.00000 q^{91} +(-1.73205 + 3.00000i) q^{92} +(-6.00000 - 10.3923i) q^{94} +(1.73205 + 3.00000i) q^{95} +(-1.00000 + 1.73205i) q^{97} -5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} - 4 q^{7}+O(q^{10}) \) 4 * q - 2 * q^4 - 4 * q^7	\( 4 q - 2 q^{4} - 4 q^{7} - 12 q^{10} + 2 q^{13} + 10 q^{16} + 8 q^{19} + 12 q^{22} + 4 q^{25} + 8 q^{28} - 16 q^{31} - 18 q^{34} - 28 q^{37} - 6 q^{40} - 4 q^{43} + 24 q^{46} + 6 q^{49} + 2 q^{52} + 24 q^{55} - 6 q^{58} + 14 q^{61} + 4 q^{64} + 20 q^{67} + 12 q^{70} - 28 q^{73} - 4 q^{76} - 4 q^{79} - 48 q^{82} + 18 q^{85} - 12 q^{88} - 8 q^{91} - 24 q^{94} - 4 q^{97}+O(q^{100}) \) 4 * q - 2 * q^4 - 4 * q^7 - 12 * q^10 + 2 * q^13 + 10 * q^16 + 8 * q^19 + 12 * q^22 + 4 * q^25 + 8 * q^28 - 16 * q^31 - 18 * q^34 - 28 * q^37 - 6 * q^40 - 4 * q^43 + 24 * q^46 + 6 * q^49 + 2 * q^52 + 24 * q^55 - 6 * q^58 + 14 * q^61 + 4 * q^64 + 20 * q^67 + 12 * q^70 - 28 * q^73 - 4 * q^76 - 4 * q^79 - 48 * q^82 + 18 * q^85 - 12 * q^88 - 8 * q^91 - 24 * q^94 - 4 * q^97

Introduction

L-functions

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