LMFDB - Embedded newform 2028.2.q.a.1837.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,2,Mod(361,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("2028.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$16.1936615299$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1837.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2028.1837
Dual form			2028.2.q.a.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} -1.73205i q^{5} +(-3.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{3} -1.73205i q^{5} +(-3.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} +(-1.50000 - 0.866025i) q^{15} +(-1.50000 - 2.59808i) q^{17} +(3.00000 - 1.73205i) q^{19} +3.46410i q^{21} +(-3.00000 + 5.19615i) q^{23} +2.00000 q^{25} -1.00000 q^{27} +(-4.50000 + 7.79423i) q^{29} +(-3.00000 + 1.73205i) q^{33} +(3.00000 + 5.19615i) q^{35} +(4.50000 + 2.59808i) q^{37} +(7.50000 + 4.33013i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(-1.50000 + 0.866025i) q^{45} +3.46410i q^{47} +(2.50000 - 4.33013i) q^{49} -3.00000 q^{51} -9.00000 q^{53} +(-3.00000 + 5.19615i) q^{55} -3.46410i q^{57} +(-12.0000 + 6.92820i) q^{59} +(5.50000 + 9.52628i) q^{61} +(3.00000 + 1.73205i) q^{63} +(-9.00000 - 5.19615i) q^{67} +(3.00000 + 5.19615i) q^{69} +(-9.00000 + 5.19615i) q^{71} +5.19615i q^{73} +(1.00000 - 1.73205i) q^{75} +12.0000 q^{77} -8.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} -3.46410i q^{83} +(-4.50000 + 2.59808i) q^{85} +(4.50000 + 7.79423i) q^{87} +(6.00000 + 3.46410i) q^{89} +(-3.00000 - 5.19615i) q^{95} +(-6.00000 + 3.46410i) q^{97} +3.46410i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 6 q^{7} - q^{9}+O(q^{10}) $ 2 * q + q^3 - 6 * q^7 - q^9	$ 2 q + q^{3} - 6 q^{7} - q^{9} - 6 q^{11} - 3 q^{15} - 3 q^{17} + 6 q^{19} - 6 q^{23} + 4 q^{25} - 2 q^{27} - 9 q^{29} - 6 q^{33} + 6 q^{35} + 9 q^{37} + 15 q^{41} - 2 q^{43} - 3 q^{45} + 5 q^{49} - 6 q^{51} - 18 q^{53} - 6 q^{55} - 24 q^{59} + 11 q^{61} + 6 q^{63} - 18 q^{67} + 6 q^{69} - 18 q^{71} + 2 q^{75} + 24 q^{77} - 16 q^{79} - q^{81} - 9 q^{85} + 9 q^{87} + 12 q^{89} - 6 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + q^3 - 6 * q^7 - q^9 - 6 * q^11 - 3 * q^15 - 3 * q^17 + 6 * q^19 - 6 * q^23 + 4 * q^25 - 2 * q^27 - 9 * q^29 - 6 * q^33 + 6 * q^35 + 9 * q^37 + 15 * q^41 - 2 * q^43 - 3 * q^45 + 5 * q^49 - 6 * q^51 - 18 * q^53 - 6 * q^55 - 24 * q^59 + 11 * q^61 + 6 * q^63 - 18 * q^67 + 6 * q^69 - 18 * q^71 + 2 * q^75 + 24 * q^77 - 16 * q^79 - q^81 - 9 * q^85 + 9 * q^87 + 12 * q^89 - 6 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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			0
$2$		0			0
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$4$		0			0
$5$		−	1.73205i		−	0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$		−	1.73205i		−	0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$		0			0
$7$	−3.00000	+	1.73205i	−1.13389	+	0.654654i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−3.00000	+	1.73205i	−1.13389	+	0.654654i	−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.0258656	−	0.999665i	$-0.508234\pi$
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.878668	+	0.477432i	$0.841568\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$	−1.50000	−	0.866025i	−0.387298	−	0.223607i
$16$		0			0
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		0			0
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	0.802955	+	0.596040i	$0.203260\pi$
$20$		0			0
$21$			3.46410i			0.755929i
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	2.00000			0.400000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	0.0578882	+	0.998323i	$0.481563\pi$
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	−0.893517	+	0.449029i	$0.851770\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	−3.00000	+	1.73205i	−0.522233	+	0.301511i
$34$		0			0
$35$	3.00000	+	5.19615i	0.507093	+	0.878310i
$36$		0			0
$37$	4.50000	+	2.59808i	0.739795	+	0.427121i	0.821995	−	0.569495i	$-0.192861\pi$
$37$	4.50000	+	2.59808i	0.739795	+	0.427121i	−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.50000	+	4.33013i	1.17130	+	0.676252i	0.953987	−	0.299849i	$-0.0969363\pi$
$41$	7.50000	+	4.33013i	1.17130	+	0.676252i	0.217317	+	0.976101i	$0.430270\pi$
$42$		0			0
$43$	−1.00000	−	1.73205i	−0.152499	−	0.264135i	0.779647	−	0.626219i	$-0.215399\pi$
$43$	−1.00000	−	1.73205i	−0.152499	−	0.264135i	−0.932145	+	0.362084i	$0.882065\pi$
$44$		0			0
$45$	−1.50000	+	0.866025i	−0.223607	+	0.129099i
$46$		0			0
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$		0			0
$49$	2.50000	−	4.33013i	0.357143	−	0.618590i
$50$		0			0
$51$	−3.00000			−0.420084
$52$		0			0
$53$	−9.00000			−1.23625			−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000			−1.23625			−0.618123	+	0.786082i	$0.712106\pi$
$54$		0			0
$55$	−3.00000	+	5.19615i	−0.404520	+	0.700649i
$56$		0			0
$57$		−	3.46410i		−	0.458831i
$58$		0			0
$59$	−12.0000	+	6.92820i	−1.56227	+	0.901975i	−0.565240	+	0.824927i	$0.691216\pi$
$59$	−12.0000	+	6.92820i	−1.56227	+	0.901975i	−0.997027	+	0.0770484i	$0.975450\pi$
$60$		0			0
$61$	5.50000	+	9.52628i	0.704203	+	1.21972i	0.966978	+	0.254858i	$0.0820288\pi$
$61$	5.50000	+	9.52628i	0.704203	+	1.21972i	−0.262776	+	0.964857i	$0.584638\pi$
$62$		0			0
$63$	3.00000	+	1.73205i	0.377964	+	0.218218i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−9.00000	−	5.19615i	−1.09952	−	0.634811i	−0.163429	−	0.986555i	$-0.552255\pi$
$67$	−9.00000	−	5.19615i	−1.09952	−	0.634811i	−0.936096	+	0.351744i	$0.885589\pi$
$68$		0			0
$69$	3.00000	+	5.19615i	0.361158	+	0.625543i
$70$		0			0
$71$	−9.00000	+	5.19615i	−1.06810	+	0.616670i	−0.927663	−	0.373419i	$-0.878185\pi$
$71$	−9.00000	+	5.19615i	−1.06810	+	0.616670i	−0.140441	+	0.990089i	$0.544852\pi$
$72$		0			0
$73$			5.19615i			0.608164i	0.952646	+	0.304082i	$0.0983496\pi$
$73$			5.19615i			0.608164i	−0.952646	+	0.304082i	$0.901650\pi$
$74$		0			0
$75$	1.00000	−	1.73205i	0.115470	−	0.200000i
$76$		0			0
$77$	12.0000			1.36753
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		−	3.46410i		−	0.380235i	−0.981761	−	0.190117i	$-0.939113\pi$
$83$		−	3.46410i		−	0.380235i	0.981761	−	0.190117i	$-0.0608868\pi$
$84$		0			0
$85$	−4.50000	+	2.59808i	−0.488094	+	0.281801i
$86$		0			0
$87$	4.50000	+	7.79423i	0.482451	+	0.835629i
$88$		0			0
$89$	6.00000	+	3.46410i	0.635999	+	0.367194i	0.783072	−	0.621932i	$-0.213652\pi$
$89$	6.00000	+	3.46410i	0.635999	+	0.367194i	−0.147073	+	0.989126i	$0.546985\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.00000	−	5.19615i	−0.307794	−	0.533114i
$96$		0			0
$97$	−6.00000	+	3.46410i	−0.609208	+	0.351726i	−0.772655	−	0.634826i	$-0.781072\pi$
$97$	−6.00000	+	3.46410i	−0.609208	+	0.351726i	0.163448	+	0.986552i	$0.447739\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	−0.930953	−	0.365140i	$-0.881021\pi$
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	0.781697	+	0.623658i	$0.214354\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$		0			0
$105$	6.00000			0.585540
$106$		0			0
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.00813215	−	0.999967i	$-0.497411\pi$
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.861931	−	0.507026i	$-0.169255\pi$
$108$		0			0
$109$		−	13.8564i		−	1.32720i	−0.748086	−	0.663602i	$-0.769027\pi$
$109$		−	13.8564i		−	1.32720i	0.748086	−	0.663602i	$-0.230973\pi$
$110$		0			0
$111$	4.50000	−	2.59808i	0.427121	−	0.246598i
$112$		0			0
$113$	−1.50000	−	2.59808i	−0.141108	−	0.244406i	0.786806	−	0.617200i	$-0.211733\pi$
$113$	−1.50000	−	2.59808i	−0.141108	−	0.244406i	−0.927914	+	0.372794i	$0.878400\pi$
$114$		0			0
$115$	9.00000	+	5.19615i	0.839254	+	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.00000	+	5.19615i	0.825029	+	0.476331i
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$		0			0
$123$	7.50000	−	4.33013i	0.676252	−	0.390434i
$124$		0			0
$125$		−	12.1244i		−	1.08444i
$126$		0			0
$127$	−8.00000	+	13.8564i	−0.709885	+	1.22956i	0.255014	+	0.966937i	$0.417920\pi$
$127$	−8.00000	+	13.8564i	−0.709885	+	1.22956i	−0.964899	+	0.262620i	$0.915413\pi$
$128$		0			0
$129$	−2.00000			−0.176090
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	−6.00000	+	10.3923i	−0.520266	+	0.901127i
$134$		0			0
$135$			1.73205i			0.149071i
$136$		0			0
$137$	−13.5000	+	7.79423i	−1.15338	+	0.665906i	−0.949709	−	0.313133i	$-0.898621\pi$
$137$	−13.5000	+	7.79423i	−1.15338	+	0.665906i	−0.203674	+	0.979039i	$0.565288\pi$
$138$		0			0
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	0.938293	−	0.345843i	$-0.112407\pi$
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	−0.768655	+	0.639664i	$0.779074\pi$
$140$		0			0
$141$	3.00000	+	1.73205i	0.252646	+	0.145865i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	13.5000	+	7.79423i	1.12111	+	0.647275i
$146$		0			0
$147$	−2.50000	−	4.33013i	−0.206197	−	0.357143i
$148$		0			0
$149$	7.50000	−	4.33013i	0.614424	−	0.354738i	−0.160271	−	0.987073i	$-0.551237\pi$
$149$	7.50000	−	4.33013i	0.614424	−	0.354738i	0.774695	+	0.632335i	$0.217903\pi$
$150$		0			0
$151$			17.3205i			1.40952i	0.709444	+	0.704761i	$0.248946\pi$
$151$			17.3205i			1.40952i	−0.709444	+	0.704761i	$0.751054\pi$
$152$		0			0
$153$	−1.50000	+	2.59808i	−0.121268	+	0.210042i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	23.0000			1.83560			0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000			1.83560			0.917800	+	0.397043i	$0.129964\pi$
$158$		0			0
$159$	−4.50000	+	7.79423i	−0.356873	+	0.618123i
$160$		0			0
$161$		−	20.7846i		−	1.63806i
$162$		0			0
$163$	12.0000	−	6.92820i	0.939913	−	0.542659i	0.0499796	−	0.998750i	$-0.484084\pi$
$163$	12.0000	−	6.92820i	0.939913	−	0.542659i	0.889933	+	0.456091i	$0.150751\pi$
$164$		0			0
$165$	3.00000	+	5.19615i	0.233550	+	0.404520i
$166$		0			0
$167$	−6.00000	−	3.46410i	−0.464294	−	0.268060i	0.249554	−	0.968361i	$-0.419716\pi$
$167$	−6.00000	−	3.46410i	−0.464294	−	0.268060i	−0.713848	+	0.700301i	$0.753049\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	−3.00000	−	1.73205i	−0.229416	−	0.132453i
$172$		0			0
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	−0.973670	−	0.227964i	$-0.926793\pi$
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	0.289412	−	0.957205i	$-0.406540\pi$
$174$		0			0
$175$	−6.00000	+	3.46410i	−0.453557	+	0.261861i
$176$		0			0
$177$			13.8564i			1.04151i
$178$		0			0
$179$	−9.00000	+	15.5885i	−0.672692	+	1.16514i	0.304446	+	0.952529i	$0.401529\pi$
$179$	−9.00000	+	15.5885i	−0.672692	+	1.16514i	−0.977138	+	0.212607i	$0.931805\pi$
$180$		0			0
$181$	−13.0000			−0.966282			−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000			−0.966282			−0.483141	+	0.875542i	$0.660504\pi$
$182$		0			0
$183$	11.0000			0.813143
$184$		0			0
$185$	4.50000	−	7.79423i	0.330847	−	0.573043i
$186$		0			0
$187$			10.3923i			0.759961i
$188$		0			0
$189$	3.00000	−	1.73205i	0.218218	−	0.125988i
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−1.50000	−	0.866025i	−0.107972	−	0.0623379i	0.445041	−	0.895510i	$-0.353189\pi$
$193$	−1.50000	−	0.866025i	−0.107972	−	0.0623379i	−0.553014	+	0.833172i	$0.686522\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000	−	3.46410i	−0.427482	−	0.246807i	0.270791	−	0.962638i	$-0.412715\pi$
$197$	−6.00000	−	3.46410i	−0.427482	−	0.246807i	−0.698273	+	0.715831i	$0.746048\pi$
$198$		0			0
$199$	−5.00000	−	8.66025i	−0.354441	−	0.613909i	0.632581	−	0.774494i	$-0.281995\pi$
$199$	−5.00000	−	8.66025i	−0.354441	−	0.613909i	−0.987022	+	0.160585i	$0.948662\pi$
$200$		0			0
$201$	−9.00000	+	5.19615i	−0.634811	+	0.366508i
$202$		0			0
$203$		−	31.1769i		−	2.18819i
$204$		0			0
$205$	7.50000	−	12.9904i	0.523823	−	0.907288i
$206$		0			0
$207$	6.00000			0.417029
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	−0.694857	−	0.719148i	$-0.744533\pi$
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	0.970229	+	0.242190i	$0.0778659\pi$
$212$		0			0
$213$			10.3923i			0.712069i
$214$		0			0
$215$	−3.00000	+	1.73205i	−0.204598	+	0.118125i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.50000	+	2.59808i	0.304082	+	0.175562i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−12.0000	−	6.92820i	−0.803579	−	0.463947i	0.0411418	−	0.999153i	$-0.486900\pi$
$223$	−12.0000	−	6.92820i	−0.803579	−	0.463947i	−0.844721	+	0.535207i	$0.820234\pi$
$224$		0			0
$225$	−1.00000	−	1.73205i	−0.0666667	−	0.115470i
$226$		0			0
$227$	−9.00000	+	5.19615i	−0.597351	+	0.344881i	−0.767999	−	0.640451i	$-0.778747\pi$
$227$	−9.00000	+	5.19615i	−0.597351	+	0.344881i	0.170648	+	0.985332i	$0.445414\pi$
$228$		0			0
$229$			6.92820i			0.457829i	0.973447	+	0.228914i	$0.0735176\pi$
$229$			6.92820i			0.457829i	−0.973447	+	0.228914i	$0.926482\pi$
$230$		0			0
$231$	6.00000	−	10.3923i	0.394771	−	0.683763i
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	6.00000			0.391397
$236$		0			0
$237$	−4.00000	+	6.92820i	−0.259828	+	0.450035i
$238$		0			0
$239$			10.3923i			0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$			10.3923i			0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$		0			0
$241$	−22.5000	+	12.9904i	−1.44935	+	0.836784i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−22.5000	+	12.9904i	−1.44935	+	0.836784i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$		0			0
$245$	−7.50000	−	4.33013i	−0.479157	−	0.276642i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−3.00000	−	1.73205i	−0.190117	−	0.109764i
$250$		0			0
$251$	12.0000	+	20.7846i	0.757433	+	1.31191i	0.944156	+	0.329500i	$0.106880\pi$
$251$	12.0000	+	20.7846i	0.757433	+	1.31191i	−0.186722	+	0.982413i	$0.559786\pi$
$252$		0			0
$253$	18.0000	−	10.3923i	1.13165	−	0.653359i
$254$		0			0
$255$			5.19615i			0.325396i
$256$		0			0
$257$	1.50000	−	2.59808i	0.0935674	−	0.162064i	−0.815442	−	0.578838i	$-0.803506\pi$
$257$	1.50000	−	2.59808i	0.0935674	−	0.162064i	0.909010	+	0.416775i	$0.136840\pi$
$258$		0			0
$259$	−18.0000			−1.11847
$260$		0			0
$261$	9.00000			0.557086
$262$		0			0
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	−0.943572	−	0.331166i	$-0.892558\pi$
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	0.758585	+	0.651575i	$0.225891\pi$
$264$		0			0
$265$			15.5885i			0.957591i
$266$		0			0
$267$	6.00000	−	3.46410i	0.367194	−	0.212000i
$268$		0			0
$269$	−15.0000	−	25.9808i	−0.914566	−	1.58408i	−0.807535	−	0.589819i	$-0.799199\pi$
$269$	−15.0000	−	25.9808i	−0.914566	−	1.58408i	−0.107031	−	0.994256i	$-0.534134\pi$
$270$		0			0
$271$	−6.00000	−	3.46410i	−0.364474	−	0.210429i	0.306568	−	0.951849i	$-0.400819\pi$
$271$	−6.00000	−	3.46410i	−0.364474	−	0.210429i	−0.671042	+	0.741420i	$0.734153\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.00000	−	3.46410i	−0.361814	−	0.208893i
$276$		0			0
$277$	9.50000	+	16.4545i	0.570800	+	0.988654i	0.996484	+	0.0837823i	$0.0267000\pi$
$277$	9.50000	+	16.4545i	0.570800	+	0.988654i	−0.425684	+	0.904872i	$0.639967\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	29.4449i		−	1.75653i	−0.478171	−	0.878267i	$-0.658700\pi$
$281$		−	29.4449i		−	1.75653i	0.478171	−	0.878267i	$-0.341300\pi$
$282$		0			0
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	−0.579437	−	0.815017i	$-0.696728\pi$
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	0.995544	+	0.0942988i	$0.0300609\pi$
$284$		0			0
$285$	−6.00000			−0.355409
$286$		0			0
$287$	−30.0000			−1.77084
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$			6.92820i			0.406138i
$292$		0			0
$293$	−1.50000	+	0.866025i	−0.0876309	+	0.0505937i	−0.543175	−	0.839619i	$-0.682778\pi$
$293$	−1.50000	+	0.866025i	−0.0876309	+	0.0505937i	0.455544	+	0.890213i	$0.349445\pi$
$294$		0			0
$295$	12.0000	+	20.7846i	0.698667	+	1.21013i
$296$		0			0
$297$	3.00000	+	1.73205i	0.174078	+	0.100504i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	6.00000	+	3.46410i	0.345834	+	0.199667i
$302$		0			0
$303$	1.50000	+	2.59808i	0.0861727	+	0.149256i
$304$		0			0
$305$	16.5000	−	9.52628i	0.944787	−	0.545473i
$306$		0			0
$307$			17.3205i			0.988534i	0.869310	+	0.494267i	$0.164563\pi$
$307$			17.3205i			0.988534i	−0.869310	+	0.494267i	$0.835437\pi$
$308$		0			0
$309$	7.00000	−	12.1244i	0.398216	−	0.689730i
$310$		0			0
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$		0			0
$313$	−14.0000			−0.791327			−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000			−0.791327			−0.395663	+	0.918396i	$0.629485\pi$
$314$		0			0
$315$	3.00000	−	5.19615i	0.169031	−	0.292770i
$316$		0			0
$317$			1.73205i			0.0972817i	0.998816	+	0.0486408i	$0.0154890\pi$
$317$			1.73205i			0.0972817i	−0.998816	+	0.0486408i	$0.984511\pi$
$318$		0			0
$319$	27.0000	−	15.5885i	1.51171	−	0.872786i
$320$		0			0
$321$	−9.00000	−	15.5885i	−0.502331	−	0.870063i
$322$		0			0
$323$	−9.00000	−	5.19615i	−0.500773	−	0.289122i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−12.0000	−	6.92820i	−0.663602	−	0.383131i
$328$		0			0
$329$	−6.00000	−	10.3923i	−0.330791	−	0.572946i
$330$		0			0
$331$	−12.0000	+	6.92820i	−0.659580	+	0.380808i	−0.792117	−	0.610370i	$-0.791021\pi$
$331$	−12.0000	+	6.92820i	−0.659580	+	0.380808i	0.132537	+	0.991178i	$0.457688\pi$
$332$		0			0
$333$		−	5.19615i		−	0.284747i
$334$		0			0
$335$	−9.00000	+	15.5885i	−0.491723	+	0.851688i
$336$		0			0
$337$	−23.0000			−1.25289			−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000			−1.25289			−0.626445	+	0.779466i	$0.715491\pi$
$338$		0			0
$339$	−3.00000			−0.162938
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$		0			0
$345$	9.00000	−	5.19615i	0.484544	−	0.279751i
$346$		0			0
$347$	−9.00000	−	15.5885i	−0.483145	−	0.836832i	0.516667	−	0.856186i	$-0.327172\pi$
$347$	−9.00000	−	15.5885i	−0.483145	−	0.836832i	−0.999813	+	0.0193540i	$0.993839\pi$
$348$		0			0
$349$	12.0000	+	6.92820i	0.642345	+	0.370858i	0.785517	−	0.618840i	$-0.212397\pi$
$349$	12.0000	+	6.92820i	0.642345	+	0.370858i	−0.143172	+	0.989698i	$0.545730\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	22.5000	+	12.9904i	1.19755	+	0.691408i	0.960009	−	0.279968i	$-0.0903240\pi$
$353$	22.5000	+	12.9904i	1.19755	+	0.691408i	0.237545	+	0.971377i	$0.423657\pi$
$354$		0			0
$355$	9.00000	+	15.5885i	0.477670	+	0.827349i
$356$		0			0
$357$	9.00000	−	5.19615i	0.476331	−	0.275010i
$358$		0			0
$359$		−	3.46410i		−	0.182828i	−0.995813	−	0.0914141i	$-0.970861\pi$
$359$		−	3.46410i		−	0.182828i	0.995813	−	0.0914141i	$-0.0291387\pi$
$360$		0			0
$361$	−3.50000	+	6.06218i	−0.184211	+	0.319062i
$362$		0			0
$363$	1.00000			0.0524864
$364$		0			0
$365$	9.00000			0.471082
$366$		0			0
$367$	−1.00000	+	1.73205i	−0.0521996	+	0.0904123i	−0.890945	−	0.454112i	$-0.849957\pi$
$367$	−1.00000	+	1.73205i	−0.0521996	+	0.0904123i	0.838745	+	0.544524i	$0.183290\pi$
$368$		0			0
$369$		−	8.66025i		−	0.450835i
$370$		0			0
$371$	27.0000	−	15.5885i	1.40177	−	0.809312i
$372$		0			0
$373$	−15.5000	−	26.8468i	−0.802560	−	1.39007i	−0.917926	−	0.396751i	$-0.870138\pi$
$373$	−15.5000	−	26.8468i	−0.802560	−	1.39007i	0.115367	−	0.993323i	$-0.463196\pi$
$374$		0			0
$375$	−10.5000	−	6.06218i	−0.542218	−	0.313050i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−12.0000	−	6.92820i	−0.616399	−	0.355878i	0.159067	−	0.987268i	$-0.449151\pi$
$379$	−12.0000	−	6.92820i	−0.616399	−	0.355878i	−0.775466	+	0.631390i	$0.782485\pi$
$380$		0			0
$381$	8.00000	+	13.8564i	0.409852	+	0.709885i
$382$		0			0
$383$	−12.0000	+	6.92820i	−0.613171	+	0.354015i	−0.774206	−	0.632934i	$-0.781850\pi$
$383$	−12.0000	+	6.92820i	−0.613171	+	0.354015i	0.161034	+	0.986949i	$0.448517\pi$
$384$		0			0
$385$		−	20.7846i		−	1.05928i
$386$		0			0
$387$	−1.00000	+	1.73205i	−0.0508329	+	0.0880451i
$388$		0			0
$389$	21.0000			1.06474			0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000			1.06474			0.532371	+	0.846511i	$0.321301\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$	−6.00000	+	10.3923i	−0.302660	+	0.524222i
$394$		0			0
$395$			13.8564i			0.697191i
$396$		0			0
$397$	24.0000	−	13.8564i	1.20453	−	0.695433i	0.242967	−	0.970034i	$-0.421879\pi$
$397$	24.0000	−	13.8564i	1.20453	−	0.695433i	0.961558	+	0.274601i	$0.0885459\pi$
$398$		0			0
$399$	6.00000	+	10.3923i	0.300376	+	0.520266i
$400$		0			0
$401$	−31.5000	−	18.1865i	−1.57303	−	0.908192i	−0.995794	−	0.0916181i	$-0.970796\pi$
$401$	−31.5000	−	18.1865i	−1.57303	−	0.908192i	−0.577241	−	0.816574i	$-0.695871\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.50000	+	0.866025i	0.0745356	+	0.0430331i
$406$		0			0
$407$	−9.00000	−	15.5885i	−0.446113	−	0.772691i
$408$		0			0
$409$	10.5000	−	6.06218i	0.519192	−	0.299755i	−0.217412	−	0.976080i	$-0.569762\pi$
$409$	10.5000	−	6.06218i	0.519192	−	0.299755i	0.736604	+	0.676324i	$0.236428\pi$
$410$		0			0
$411$			15.5885i			0.768922i
$412$		0			0
$413$	24.0000	−	41.5692i	1.18096	−	2.04549i
$414$		0			0
$415$	−6.00000			−0.294528
$416$		0			0
$417$	4.00000			0.195881
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$			12.1244i			0.590905i	0.955357	+	0.295452i	$0.0954704\pi$
$421$			12.1244i			0.590905i	−0.955357	+	0.295452i	$0.904530\pi$
$422$		0			0
$423$	3.00000	−	1.73205i	0.145865	−	0.0842152i
$424$		0			0
$425$	−3.00000	−	5.19615i	−0.145521	−	0.252050i
$426$		0			0
$427$	−33.0000	−	19.0526i	−1.59698	−	0.922018i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	27.0000	+	15.5885i	1.30054	+	0.750870i	0.980497	−	0.196532i	$-0.0629680\pi$
$431$	27.0000	+	15.5885i	1.30054	+	0.750870i	0.320047	+	0.947402i	$0.396301\pi$
$432$		0			0
$433$	14.5000	+	25.1147i	0.696826	+	1.20694i	0.969561	+	0.244848i	$0.0787382\pi$
$433$	14.5000	+	25.1147i	0.696826	+	1.20694i	−0.272736	+	0.962089i	$0.587929\pi$
$434$		0			0
$435$	13.5000	−	7.79423i	0.647275	−	0.373705i
$436$		0			0
$437$			20.7846i			0.994263i
$438$		0			0
$439$	5.00000	−	8.66025i	0.238637	−	0.413331i	−0.721686	−	0.692220i	$-0.756633\pi$
$439$	5.00000	−	8.66025i	0.238637	−	0.413331i	0.960323	+	0.278889i	$0.0899661\pi$
$440$		0			0
$441$	−5.00000			−0.238095
$442$		0			0
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$		0			0
$445$	6.00000	−	10.3923i	0.284427	−	0.492642i
$446$		0			0
$447$		−	8.66025i		−	0.409616i
$448$		0			0
$449$	6.00000	−	3.46410i	0.283158	−	0.163481i	−0.351694	−	0.936115i	$-0.614394\pi$
$449$	6.00000	−	3.46410i	0.283158	−	0.163481i	0.634852	+	0.772634i	$0.281061\pi$
$450$		0			0
$451$	−15.0000	−	25.9808i	−0.706322	−	1.22339i
$452$		0			0
$453$	15.0000	+	8.66025i	0.704761	+	0.406894i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.50000	−	4.33013i	−0.350835	−	0.202555i	0.314218	−	0.949351i	$-0.398258\pi$
$457$	−7.50000	−	4.33013i	−0.350835	−	0.202555i	−0.665053	+	0.746796i	$0.731591\pi$
$458$		0			0
$459$	1.50000	+	2.59808i	0.0700140	+	0.121268i
$460$		0			0
$461$	−4.50000	+	2.59808i	−0.209586	+	0.121004i	−0.601119	−	0.799160i	$-0.705278\pi$
$461$	−4.50000	+	2.59808i	−0.209586	+	0.121004i	0.391533	+	0.920164i	$0.371945\pi$
$462$		0			0
$463$		−	3.46410i		−	0.160990i	−0.996755	−	0.0804952i	$-0.974350\pi$
$463$		−	3.46410i		−	0.160990i	0.996755	−	0.0804952i	$-0.0256502\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.00000			−0.277647			−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000			−0.277647			−0.138823	+	0.990317i	$0.544332\pi$
$468$		0			0
$469$	36.0000			1.66233
$470$		0			0
$471$	11.5000	−	19.9186i	0.529892	−	0.917800i
$472$		0			0
$473$			6.92820i			0.318559i
$474$		0			0
$475$	6.00000	−	3.46410i	0.275299	−	0.158944i
$476$		0			0
$477$	4.50000	+	7.79423i	0.206041	+	0.356873i
$478$		0			0
$479$	−6.00000	−	3.46410i	−0.274147	−	0.158279i	0.356624	−	0.934248i	$-0.383928\pi$
$479$	−6.00000	−	3.46410i	−0.274147	−	0.158279i	−0.630771	+	0.775969i	$0.717261\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−18.0000	−	10.3923i	−0.819028	−	0.472866i
$484$		0			0
$485$	6.00000	+	10.3923i	0.272446	+	0.471890i
$486$		0			0
$487$	15.0000	−	8.66025i	0.679715	−	0.392434i	−0.120033	−	0.992770i	$-0.538300\pi$
$487$	15.0000	−	8.66025i	0.679715	−	0.392434i	0.799748	+	0.600336i	$0.204967\pi$
$488$		0			0
$489$		−	13.8564i		−	0.626608i
$490$		0			0
$491$	−15.0000	+	25.9808i	−0.676941	+	1.17250i	0.298957	+	0.954267i	$0.403361\pi$
$491$	−15.0000	+	25.9808i	−0.676941	+	1.17250i	−0.975898	+	0.218229i	$0.929972\pi$
$492$		0			0
$493$	27.0000			1.21602
$494$		0			0
$495$	6.00000			0.269680
$496$		0			0
$497$	18.0000	−	31.1769i	0.807410	−	1.39848i
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	−6.00000	+	3.46410i	−0.268060	+	0.154765i
$502$		0			0
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	0.791361	−	0.611349i	$-0.209373\pi$
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	−0.925124	+	0.379664i	$0.876040\pi$
$504$		0			0
$505$	4.50000	+	2.59808i	0.200247	+	0.115613i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.50000	−	0.866025i	−0.0664863	−	0.0383859i	0.466388	−	0.884580i	$-0.345555\pi$
$509$	−1.50000	−	0.866025i	−0.0664863	−	0.0383859i	−0.532875	+	0.846194i	$0.678888\pi$
$510$		0			0
$511$	−9.00000	−	15.5885i	−0.398137	−	0.689593i
$512$		0			0
$513$	−3.00000	+	1.73205i	−0.132453	+	0.0764719i
$514$		0			0
$515$		−	24.2487i		−	1.06853i
$516$		0			0
$517$	6.00000	−	10.3923i	0.263880	−	0.457053i
$518$		0			0
$519$	−18.0000			−0.790112
$520$		0			0
$521$	−9.00000			−0.394297			−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000			−0.394297			−0.197149	+	0.980374i	$0.563168\pi$
$522$		0			0
$523$	5.00000	−	8.66025i	0.218635	−	0.378686i	−0.735756	−	0.677247i	$-0.763173\pi$
$523$	5.00000	−	8.66025i	0.218635	−	0.378686i	0.954391	+	0.298560i	$0.0965063\pi$
$524$		0			0
$525$			6.92820i			0.302372i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$	12.0000	+	6.92820i	0.520756	+	0.300658i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−27.0000	−	15.5885i	−1.16731	−	0.673948i
$536$		0			0
$537$	9.00000	+	15.5885i	0.388379	+	0.672692i
$538$		0			0
$539$	−15.0000	+	8.66025i	−0.646096	+	0.373024i
$540$		0			0
$541$		−	1.73205i		−	0.0744667i	−0.999307	−	0.0372333i	$-0.988146\pi$
$541$		−	1.73205i		−	0.0744667i	0.999307	−	0.0372333i	$-0.0118545\pi$
$542$		0			0
$543$	−6.50000	+	11.2583i	−0.278942	+	0.483141i
$544$		0			0
$545$	−24.0000			−1.02805
$546$		0			0
$547$	−34.0000			−1.45374			−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000			−1.45374			−0.726868	+	0.686778i	$0.759025\pi$
$548$		0			0
$549$	5.50000	−	9.52628i	0.234734	−	0.406572i
$550$		0			0
$551$			31.1769i			1.32818i
$552$		0			0
$553$	24.0000	−	13.8564i	1.02058	−	0.589234i
$554$		0			0
$555$	−4.50000	−	7.79423i	−0.191014	−	0.330847i
$556$		0			0
$557$	−1.50000	−	0.866025i	−0.0635570	−	0.0366947i	0.467885	−	0.883789i	$-0.345016\pi$
$557$	−1.50000	−	0.866025i	−0.0635570	−	0.0366947i	−0.531442	+	0.847095i	$0.678350\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	9.00000	+	5.19615i	0.379980	+	0.219382i
$562$		0			0
$563$	−12.0000	−	20.7846i	−0.505740	−	0.875967i	−0.999978	−	0.00664037i	$-0.997886\pi$
$563$	−12.0000	−	20.7846i	−0.505740	−	0.875967i	0.494238	−	0.869326i	$-0.335447\pi$
$564$		0			0
$565$	−4.50000	+	2.59808i	−0.189316	+	0.109302i
$566$		0			0
$567$		−	3.46410i		−	0.145479i
$568$		0			0
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−26.0000			−1.08807			−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000			−1.08807			−0.544033	+	0.839064i	$0.683103\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.00000	+	10.3923i	−0.250217	+	0.433389i
$576$		0			0
$577$		−	8.66025i		−	0.360531i	−0.983618	−	0.180266i	$-0.942304\pi$
$577$		−	8.66025i		−	0.360531i	0.983618	−	0.180266i	$-0.0576957\pi$
$578$		0			0
$579$	−1.50000	+	0.866025i	−0.0623379	+	0.0359908i
$580$		0			0
$581$	6.00000	+	10.3923i	0.248922	+	0.431145i
$582$		0			0
$583$	27.0000	+	15.5885i	1.11823	+	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.0000	−	10.3923i	−0.742940	−	0.428936i	0.0801976	−	0.996779i	$-0.474445\pi$
$587$	−18.0000	−	10.3923i	−0.742940	−	0.428936i	−0.823137	+	0.567843i	$0.807778\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−6.00000	+	3.46410i	−0.246807	+	0.142494i
$592$		0			0
$593$		−	12.1244i		−	0.497888i	−0.968518	−	0.248944i	$-0.919917\pi$
$593$		−	12.1244i		−	0.497888i	0.968518	−	0.248944i	$-0.0800834\pi$
$594$		0			0
$595$	9.00000	−	15.5885i	0.368964	−	0.639064i
$596$		0			0
$597$	−10.0000			−0.409273
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	−18.5000	+	32.0429i	−0.754631	+	1.30706i	0.190927	+	0.981604i	$0.438851\pi$
$601$	−18.5000	+	32.0429i	−0.754631	+	1.30706i	−0.945558	+	0.325455i	$0.894483\pi$
$602$		0			0
$603$			10.3923i			0.423207i
$604$		0			0
$605$	1.50000	−	0.866025i	0.0609837	−	0.0352089i
$606$		0			0
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	0.773358	−	0.633970i	$-0.218576\pi$
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	−0.935713	+	0.352763i	$0.885242\pi$
$608$		0			0
$609$	−27.0000	−	15.5885i	−1.09410	−	0.631676i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	4.50000	+	2.59808i	0.181753	+	0.104935i	0.588116	−	0.808776i	$-0.299870\pi$
$613$	4.50000	+	2.59808i	0.181753	+	0.104935i	−0.406363	+	0.913712i	$0.633203\pi$
$614$		0			0
$615$	−7.50000	−	12.9904i	−0.302429	−	0.523823i
$616$		0			0
$617$	22.5000	−	12.9904i	0.905816	−	0.522973i	0.0267333	−	0.999643i	$-0.491490\pi$
$617$	22.5000	−	12.9904i	0.905816	−	0.522973i	0.879083	+	0.476670i	$0.158156\pi$
$618$		0			0
$619$		−	13.8564i		−	0.556936i	−0.960446	−	0.278468i	$-0.910173\pi$
$619$		−	13.8564i		−	0.556936i	0.960446	−	0.278468i	$-0.0898266\pi$
$620$		0			0
$621$	3.00000	−	5.19615i	0.120386	−	0.208514i
$622$		0			0
$623$	−24.0000			−0.961540
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	−6.00000	+	10.3923i	−0.239617	+	0.415029i
$628$		0			0
$629$		−	15.5885i		−	0.621552i
$630$		0			0
$631$	−24.0000	+	13.8564i	−0.955425	+	0.551615i	−0.894762	−	0.446543i	$-0.852655\pi$
$631$	−24.0000	+	13.8564i	−0.955425	+	0.551615i	−0.0606630	+	0.998158i	$0.519321\pi$
$632$		0			0
$633$	−4.00000	−	6.92820i	−0.158986	−	0.275371i
$634$		0			0
$635$	24.0000	+	13.8564i	0.952411	+	0.549875i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	9.00000	+	5.19615i	0.356034	+	0.205557i
$640$		0			0
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	0.995400	−	0.0958109i	$-0.0305444\pi$
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	−0.580674	+	0.814136i	$0.697211\pi$
$642$		0			0
$643$	6.00000	−	3.46410i	0.236617	−	0.136611i	−0.377004	−	0.926212i	$-0.623046\pi$
$643$	6.00000	−	3.46410i	0.236617	−	0.136611i	0.613621	+	0.789601i	$0.289712\pi$
$644$		0			0
$645$			3.46410i			0.136399i
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$	48.0000			1.88416
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	−0.918736	−	0.394872i	$-0.870789\pi$
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	0.801337	+	0.598213i	$0.204122\pi$
$654$		0			0
$655$			20.7846i			0.812122i
$656$		0			0
$657$	4.50000	−	2.59808i	0.175562	−	0.101361i
$658$		0			0
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	0.725175	−	0.688565i	$-0.241759\pi$
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	−0.958902	+	0.283738i	$0.908425\pi$
$660$		0			0
$661$	10.5000	+	6.06218i	0.408403	+	0.235791i	0.690103	−	0.723711i	$-0.257565\pi$
$661$	10.5000	+	6.06218i	0.408403	+	0.235791i	−0.281701	+	0.959502i	$0.590898\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18.0000	+	10.3923i	0.698010	+	0.402996i
$666$		0			0
$667$	−27.0000	−	46.7654i	−1.04544	−	1.81076i
$668$		0			0
$669$	−12.0000	+	6.92820i	−0.463947	+	0.267860i
$670$		0			0
$671$		−	38.1051i		−	1.47103i
$672$		0			0
$673$	−2.50000	+	4.33013i	−0.0963679	+	0.166914i	−0.910179	−	0.414216i	$-0.864056\pi$
$673$	−2.50000	+	4.33013i	−0.0963679	+	0.166914i	0.813811	+	0.581130i	$0.197389\pi$
$674$		0			0
$675$	−2.00000			−0.0769800
$676$		0			0
$677$	−30.0000			−1.15299			−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000			−1.15299			−0.576497	+	0.817099i	$0.695581\pi$
$678$		0			0
$679$	12.0000	−	20.7846i	0.460518	−	0.797640i
$680$		0			0
$681$			10.3923i			0.398234i
$682$		0			0
$683$	12.0000	−	6.92820i	0.459167	−	0.265100i	−0.252527	−	0.967590i	$-0.581262\pi$
$683$	12.0000	−	6.92820i	0.459167	−	0.265100i	0.711694	+	0.702490i	$0.247928\pi$
$684$		0			0
$685$	13.5000	+	23.3827i	0.515808	+	0.893407i
$686$		0			0
$687$	6.00000	+	3.46410i	0.228914	+	0.132164i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	9.00000	+	5.19615i	0.342376	+	0.197671i	0.661322	−	0.750102i	$-0.269996\pi$
$691$	9.00000	+	5.19615i	0.342376	+	0.197671i	−0.318946	+	0.947773i	$0.603329\pi$
$692$		0			0
$693$	−6.00000	−	10.3923i	−0.227921	−	0.394771i
$694$		0			0
$695$	6.00000	−	3.46410i	0.227593	−	0.131401i
$696$		0			0
$697$		−	25.9808i		−	0.984092i
$698$		0			0
$699$	9.00000	−	15.5885i	0.340411	−	0.589610i
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	18.0000			0.678883
$704$		0			0
$705$	3.00000	−	5.19615i	0.112987	−	0.195698i
$706$		0			0
$707$		−	10.3923i		−	0.390843i
$708$		0			0
$709$	−13.5000	+	7.79423i	−0.507003	+	0.292718i	−0.731601	−	0.681733i	$-0.761227\pi$
$709$	−13.5000	+	7.79423i	−0.507003	+	0.292718i	0.224598	+	0.974452i	$0.427893\pi$
$710$		0			0
$711$	4.00000	+	6.92820i	0.150012	+	0.259828i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	9.00000	+	5.19615i	0.336111	+	0.194054i
$718$		0			0
$719$	18.0000	+	31.1769i	0.671287	+	1.16270i	0.977539	+	0.210752i	$0.0675914\pi$
$719$	18.0000	+	31.1769i	0.671287	+	1.16270i	−0.306253	+	0.951950i	$0.599075\pi$
$720$		0			0
$721$	−42.0000	+	24.2487i	−1.56416	+	0.903069i
$722$		0			0
$723$			25.9808i			0.966235i
$724$		0			0
$725$	−9.00000	+	15.5885i	−0.334252	+	0.578941i
$726$		0			0
$727$	10.0000			0.370879			0.185440	−	0.982656i	$-0.440629\pi$
$727$	10.0000			0.370879			0.185440	+	0.982656i	$0.440629\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−3.00000	+	5.19615i	−0.110959	+	0.192187i
$732$		0			0
$733$			36.3731i			1.34347i	0.740792	+	0.671735i	$0.234451\pi$
$733$			36.3731i			1.34347i	−0.740792	+	0.671735i	$0.765549\pi$
$734$		0			0
$735$	−7.50000	+	4.33013i	−0.276642	+	0.159719i
$736$		0			0
$737$	18.0000	+	31.1769i	0.663039	+	1.14842i
$738$		0			0
$739$	−24.0000	−	13.8564i	−0.882854	−	0.509716i	−0.0112558	−	0.999937i	$-0.503583\pi$
$739$	−24.0000	−	13.8564i	−0.882854	−	0.509716i	−0.871598	+	0.490221i	$0.836916\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−36.0000	−	20.7846i	−1.32071	−	0.762513i	−0.336870	−	0.941551i	$-0.609368\pi$
$743$	−36.0000	−	20.7846i	−1.32071	−	0.762513i	−0.983842	+	0.179038i	$0.942702\pi$
$744$		0			0
$745$	−7.50000	−	12.9904i	−0.274779	−	0.475931i
$746$		0			0
$747$	−3.00000	+	1.73205i	−0.109764	+	0.0633724i
$748$		0			0
$749$			62.3538i			2.27836i
$750$		0			0
$751$	−13.0000	+	22.5167i	−0.474377	+	0.821645i	−0.999570	−	0.0293387i	$-0.990660\pi$
$751$	−13.0000	+	22.5167i	−0.474377	+	0.821645i	0.525193	+	0.850983i	$0.323993\pi$
$752$		0			0
$753$	24.0000			0.874609
$754$		0			0
$755$	30.0000			1.09181
$756$		0			0
$757$	1.00000	−	1.73205i	0.0363456	−	0.0629525i	−0.847280	−	0.531146i	$-0.821762\pi$
$757$	1.00000	−	1.73205i	0.0363456	−	0.0629525i	0.883626	+	0.468193i	$0.155095\pi$
$758$		0			0
$759$		−	20.7846i		−	0.754434i
$760$		0			0
$761$	−18.0000	+	10.3923i	−0.652499	+	0.376721i	−0.789413	−	0.613862i	$-0.789615\pi$
$761$	−18.0000	+	10.3923i	−0.652499	+	0.376721i	0.136914	+	0.990583i	$0.456282\pi$
$762$		0			0
$763$	24.0000	+	41.5692i	0.868858	+	1.50491i
$764$		0			0
$765$	4.50000	+	2.59808i	0.162698	+	0.0939336i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	30.0000	+	17.3205i	1.08183	+	0.624593i	0.931389	−	0.364026i	$-0.118598\pi$
$769$	30.0000	+	17.3205i	1.08183	+	0.624593i	0.150439	+	0.988619i	$0.451931\pi$
$770$		0			0
$771$	−1.50000	−	2.59808i	−0.0540212	−	0.0935674i
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−9.00000	+	15.5885i	−0.322873	+	0.559233i
$778$		0			0
$779$	30.0000			1.07486
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$	4.50000	−	7.79423i	0.160817	−	0.278543i
$784$		0			0
$785$		−	39.8372i		−	1.42185i
$786$		0			0
$787$	36.0000	−	20.7846i	1.28326	−	0.740891i	0.305818	−	0.952090i	$-0.401070\pi$
$787$	36.0000	−	20.7846i	1.28326	−	0.740891i	0.977443	+	0.211199i	$0.0677367\pi$
$788$		0			0
$789$	3.00000	+	5.19615i	0.106803	+	0.184988i
$790$		0			0
$791$	9.00000	+	5.19615i	0.320003	+	0.184754i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	13.5000	+	7.79423i	0.478796	+	0.276433i
$796$		0			0
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	0.661441	−	0.749997i	$-0.269945\pi$
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	−0.980237	+	0.197826i	$0.936612\pi$
$798$		0			0
$799$	9.00000	−	5.19615i	0.318397	−	0.183827i
$800$		0			0
$801$		−	6.92820i		−	0.244796i
$802$		0			0
$803$	9.00000	−	15.5885i	0.317603	−	0.550105i
$804$		0			0
$805$	−36.0000			−1.26883
$806$		0			0
$807$	−30.0000			−1.05605
$808$		0			0
$809$	10.5000	−	18.1865i	0.369160	−	0.639404i	−0.620274	−	0.784385i	$-0.712979\pi$
$809$	10.5000	−	18.1865i	0.369160	−	0.639404i	0.989434	+	0.144981i	$0.0463120\pi$
$810$		0			0
$811$		−	6.92820i		−	0.243282i	−0.992574	−	0.121641i	$-0.961184\pi$
$811$		−	6.92820i		−	0.243282i	0.992574	−	0.121641i	$-0.0388157\pi$
$812$		0			0
$813$	−6.00000	+	3.46410i	−0.210429	+	0.121491i
$814$		0			0
$815$	−12.0000	−	20.7846i	−0.420342	−	0.728053i
$816$		0			0
$817$	−6.00000	−	3.46410i	−0.209913	−	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.0000	−	13.8564i	−0.837606	−	0.483592i	0.0188439	−	0.999822i	$-0.494001\pi$
$821$	−24.0000	−	13.8564i	−0.837606	−	0.483592i	−0.856450	+	0.516231i	$0.827335\pi$
$822$		0			0
$823$	20.0000	+	34.6410i	0.697156	+	1.20751i	0.969448	+	0.245295i	$0.0788849\pi$
$823$	20.0000	+	34.6410i	0.697156	+	1.20751i	−0.272292	+	0.962215i	$0.587782\pi$
$824$		0			0
$825$	−6.00000	+	3.46410i	−0.208893	+	0.120605i
$826$		0			0
$827$			27.7128i			0.963669i	0.876262	+	0.481834i	$0.160029\pi$
$827$			27.7128i			0.963669i	−0.876262	+	0.481834i	$0.839971\pi$
$828$		0			0
$829$	5.50000	−	9.52628i	0.191023	−	0.330861i	−0.754567	−	0.656223i	$-0.772153\pi$
$829$	5.50000	−	9.52628i	0.191023	−	0.330861i	0.945589	+	0.325362i	$0.105486\pi$
$830$		0			0
$831$	19.0000			0.659103
$832$		0			0
$833$	−15.0000			−0.519719
$834$		0			0
$835$	−6.00000	+	10.3923i	−0.207639	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36.0000	+	20.7846i	−1.24286	+	0.717564i	−0.969675	−	0.244398i	$-0.921410\pi$
$839$	−36.0000	+	20.7846i	−1.24286	+	0.717564i	−0.273183	+	0.961962i	$0.588076\pi$
$840$		0			0
$841$	−26.0000	−	45.0333i	−0.896552	−	1.55287i
$842$		0			0
$843$	−25.5000	−	14.7224i	−0.878267	−	0.507067i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−3.00000	−	1.73205i	−0.103081	−	0.0595140i
$848$		0			0
$849$	−7.00000	−	12.1244i	−0.240239	−	0.416107i
$850$		0			0
$851$	−27.0000	+	15.5885i	−0.925548	+	0.534365i
$852$		0			0
$853$			43.3013i			1.48261i	0.671170	+	0.741304i	$0.265792\pi$
$853$			43.3013i			1.48261i	−0.671170	+	0.741304i	$0.734208\pi$
$854$		0			0
$855$	−3.00000	+	5.19615i	−0.102598	+	0.177705i
$856$		0			0
$857$	−3.00000			−0.102478			−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000			−0.102478			−0.0512390	+	0.998686i	$0.516317\pi$
$858$		0			0
$859$	−14.0000			−0.477674			−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000			−0.477674			−0.238837	+	0.971060i	$0.576766\pi$
$860$		0			0
$861$	−15.0000	+	25.9808i	−0.511199	+	0.885422i
$862$		0			0
$863$		−	38.1051i		−	1.29711i	−0.761166	−	0.648557i	$-0.775373\pi$
$863$		−	38.1051i		−	1.29711i	0.761166	−	0.648557i	$-0.224627\pi$
$864$		0			0
$865$	−27.0000	+	15.5885i	−0.918028	+	0.530023i
$866$		0			0
$867$	−4.00000	−	6.92820i	−0.135847	−	0.235294i
$868$		0			0
$869$	24.0000	+	13.8564i	0.814144	+	0.470046i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	6.00000	+	3.46410i	0.203069	+	0.117242i
$874$		0			0
$875$	21.0000	+	36.3731i	0.709930	+	1.22963i
$876$		0			0
$877$	31.5000	−	18.1865i	1.06368	−	0.614116i	0.137232	−	0.990539i	$-0.456180\pi$
$877$	31.5000	−	18.1865i	1.06368	−	0.614116i	0.926448	+	0.376423i	$0.122846\pi$
$878$		0			0
$879$			1.73205i			0.0584206i
$880$		0			0
$881$	−4.50000	+	7.79423i	−0.151609	+	0.262594i	−0.931819	−	0.362923i	$-0.881779\pi$
$881$	−4.50000	+	7.79423i	−0.151609	+	0.262594i	0.780210	+	0.625517i	$0.215112\pi$
$882$		0			0
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$		0			0
$885$	24.0000			0.806751
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		−	55.4256i		−	1.85892i
$890$		0			0
$891$	3.00000	−	1.73205i	0.100504	−	0.0580259i
$892$		0			0
$893$	6.00000	+	10.3923i	0.200782	+	0.347765i
$894$		0			0
$895$	27.0000	+	15.5885i	0.902510	+	0.521065i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	13.5000	+	23.3827i	0.449750	+	0.778990i
$902$		0			0
$903$	6.00000	−	3.46410i	0.199667	−	0.115278i
$904$		0			0
$905$			22.5167i			0.748479i
$906$		0			0
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	−0.999195	−	0.0401089i	$-0.987230\pi$
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	0.534333	+	0.845274i	$0.320563\pi$
$908$		0			0
$909$	3.00000			0.0995037
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−6.00000	+	10.3923i	−0.198571	+	0.343935i
$914$		0			0
$915$		−	19.0526i		−	0.629858i
$916$		0			0
$917$	36.0000	−	20.7846i	1.18882	−	0.686368i
$918$		0			0
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	0.999475	+	0.0323936i	$0.0103130\pi$
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	−0.471684	+	0.881768i	$0.656354\pi$
$920$		0			0
$921$	15.0000	+	8.66025i	0.494267	+	0.285365i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	9.00000	+	5.19615i	0.295918	+	0.170848i
$926$		0			0
$927$	−7.00000	−	12.1244i	−0.229910	−	0.398216i
$928$		0			0
$929$	49.5000	−	28.5788i	1.62404	−	0.937641i	0.638219	−	0.769855i	$-0.279671\pi$
$929$	49.5000	−	28.5788i	1.62404	−	0.937641i	0.985823	−	0.167786i	$-0.0536619\pi$
$930$		0			0
$931$		−	17.3205i		−	0.567657i
$932$		0			0
$933$	−9.00000	+	15.5885i	−0.294647	+	0.510343i
$934$		0			0
$935$	18.0000			0.588663
$936$		0			0
$937$	−41.0000			−1.33941			−0.669706	−	0.742627i	$-0.733580\pi$
$937$	−41.0000			−1.33941			−0.669706	+	0.742627i	$0.733580\pi$
$938$		0			0
$939$	−7.00000	+	12.1244i	−0.228436	+	0.395663i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$	−45.0000	+	25.9808i	−1.46540	+	0.846050i
$944$		0			0
$945$	−3.00000	−	5.19615i	−0.0975900	−	0.169031i
$946$		0			0
$947$	12.0000	+	6.92820i	0.389948	+	0.225136i	0.682137	−	0.731224i	$-0.261051\pi$
$947$	12.0000	+	6.92820i	0.389948	+	0.225136i	−0.292190	+	0.956360i	$0.594384\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	1.50000	+	0.866025i	0.0486408	+	0.0280828i
$952$		0			0
$953$	−27.0000	−	46.7654i	−0.874616	−	1.51488i	−0.857171	−	0.515031i	$-0.827780\pi$
$953$	−27.0000	−	46.7654i	−0.874616	−	1.51488i	−0.0174443	−	0.999848i	$-0.505553\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	31.1769i		−	1.00781i
$958$		0			0
$959$	27.0000	−	46.7654i	0.871875	−	1.51013i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−18.0000			−0.580042
$964$		0			0
$965$	−1.50000	+	2.59808i	−0.0482867	+	0.0836350i
$966$		0			0
$967$		−	45.0333i		−	1.44817i	−0.689709	−	0.724087i	$-0.742261\pi$
$967$		−	45.0333i		−	1.44817i	0.689709	−	0.724087i	$-0.257739\pi$
$968$		0			0
$969$	−9.00000	+	5.19615i	−0.289122	+	0.166924i
$970$		0			0
$971$	12.0000	+	20.7846i	0.385098	+	0.667010i	0.991783	−	0.127933i	$-0.0408342\pi$
$971$	12.0000	+	20.7846i	0.385098	+	0.667010i	−0.606685	+	0.794943i	$0.707501\pi$
$972$		0			0
$973$	−12.0000	−	6.92820i	−0.384702	−	0.222108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.50000	+	4.33013i	0.239946	+	0.138533i	0.615152	−	0.788409i	$-0.289095\pi$
$977$	7.50000	+	4.33013i	0.239946	+	0.138533i	−0.375206	+	0.926942i	$0.622428\pi$
$978$		0			0
$979$	−12.0000	−	20.7846i	−0.383522	−	0.664279i
$980$		0			0
$981$	−12.0000	+	6.92820i	−0.383131	+	0.221201i
$982$		0			0
$983$			20.7846i			0.662926i	0.943468	+	0.331463i	$0.107542\pi$
$983$			20.7846i			0.662926i	−0.943468	+	0.331463i	$0.892458\pi$
$984$		0			0
$985$	−6.00000	+	10.3923i	−0.191176	+	0.331126i
$986$		0			0
$987$	−12.0000			−0.381964
$988$		0			0
$989$	12.0000			0.381578
$990$		0			0
$991$	−1.00000	+	1.73205i	−0.0317660	+	0.0550204i	−0.881471	−	0.472237i	$-0.843446\pi$
$991$	−1.00000	+	1.73205i	−0.0317660	+	0.0550204i	0.849705	+	0.527258i	$0.176780\pi$
$992$		0			0
$993$			13.8564i			0.439720i
$994$		0			0
$995$	−15.0000	+	8.66025i	−0.475532	+	0.274549i
$996$		0			0
$997$	27.5000	+	47.6314i	0.870934	+	1.50850i	0.861032	+	0.508551i	$0.169818\pi$
$997$	27.5000	+	47.6314i	0.870934	+	1.50850i	0.00990158	+	0.999951i	$0.496848\pi$
$998$		0			0
$999$	−4.50000	−	2.59808i	−0.142374	−	0.0821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.q.a.1837.1		2
13.2	odd	12		2028.2.i.h.2005.1		4
13.3	even	3		156.2.q.a.49.1	✓	2
13.4	even	6		2028.2.b.b.337.2		2
13.5	odd	4		2028.2.i.h.529.1		4
13.6	odd	12		2028.2.a.h.1.1		2
13.7	odd	12		2028.2.a.h.1.2		2
13.8	odd	4		2028.2.i.h.529.2		4
13.9	even	3		2028.2.b.b.337.1		2
13.10	even	6	inner	2028.2.q.a.361.1		2
13.11	odd	12		2028.2.i.h.2005.2		4
13.12	even	2		156.2.q.a.121.1	yes	2
39.17	odd	6		6084.2.b.c.4393.1		2
39.20	even	12		6084.2.a.u.1.1		2
39.29	odd	6		468.2.t.c.361.1		2
39.32	even	12		6084.2.a.u.1.2		2
39.35	odd	6		6084.2.b.c.4393.2		2
39.38	odd	2		468.2.t.c.433.1		2
52.3	odd	6		624.2.bv.a.49.1		2
52.7	even	12		8112.2.a.bt.1.2		2
52.19	even	12		8112.2.a.bt.1.1		2
52.51	odd	2		624.2.bv.a.433.1		2
65.3	odd	12		3900.2.bw.e.49.1		4
65.12	odd	4		3900.2.bw.e.2149.1		4
65.29	even	6		3900.2.cd.a.2701.1		2
65.38	odd	4		3900.2.bw.e.2149.2		4
65.42	odd	12		3900.2.bw.e.49.2		4
65.64	even	2		3900.2.cd.a.901.1		2
156.107	even	6		1872.2.by.b.1297.1		2
156.155	even	2		1872.2.by.b.433.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.q.a.49.1	✓	2	13.3	even	3
156.2.q.a.121.1	yes	2	13.12	even	2
468.2.t.c.361.1		2	39.29	odd	6
468.2.t.c.433.1		2	39.38	odd	2
624.2.bv.a.49.1		2	52.3	odd	6
624.2.bv.a.433.1		2	52.51	odd	2
1872.2.by.b.433.1		2	156.155	even	2
1872.2.by.b.1297.1		2	156.107	even	6
2028.2.a.h.1.1		2	13.6	odd	12
2028.2.a.h.1.2		2	13.7	odd	12
2028.2.b.b.337.1		2	13.9	even	3
2028.2.b.b.337.2		2	13.4	even	6
2028.2.i.h.529.1		4	13.5	odd	4
2028.2.i.h.529.2		4	13.8	odd	4
2028.2.i.h.2005.1		4	13.2	odd	12
2028.2.i.h.2005.2		4	13.11	odd	12
2028.2.q.a.361.1		2	13.10	even	6	inner
2028.2.q.a.1837.1		2	1.1	even	1	trivial
3900.2.bw.e.49.1		4	65.3	odd	12
3900.2.bw.e.49.2		4	65.42	odd	12
3900.2.bw.e.2149.1		4	65.12	odd	4
3900.2.bw.e.2149.2		4	65.38	odd	4
3900.2.cd.a.901.1		2	65.64	even	2
3900.2.cd.a.2701.1		2	65.29	even	6
6084.2.a.u.1.1		2	39.20	even	12
6084.2.a.u.1.2		2	39.32	even	12
6084.2.b.c.4393.1		2	39.17	odd	6
6084.2.b.c.4393.2		2	39.35	odd	6
8112.2.a.bt.1.1		2	52.19	even	12
8112.2.a.bt.1.2		2	52.7	even	12

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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} -1.73205i q^{5} +(-3.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} -1.73205i q^{5} +(-3.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} +(-1.50000 - 0.866025i) q^{15} +(-1.50000 - 2.59808i) q^{17} +(3.00000 - 1.73205i) q^{19} +3.46410i q^{21} +(-3.00000 + 5.19615i) q^{23} +2.00000 q^{25} -1.00000 q^{27} +(-4.50000 + 7.79423i) q^{29} +(-3.00000 + 1.73205i) q^{33} +(3.00000 + 5.19615i) q^{35} +(4.50000 + 2.59808i) q^{37} +(7.50000 + 4.33013i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(-1.50000 + 0.866025i) q^{45} +3.46410i q^{47} +(2.50000 - 4.33013i) q^{49} -3.00000 q^{51} -9.00000 q^{53} +(-3.00000 + 5.19615i) q^{55} -3.46410i q^{57} +(-12.0000 + 6.92820i) q^{59} +(5.50000 + 9.52628i) q^{61} +(3.00000 + 1.73205i) q^{63} +(-9.00000 - 5.19615i) q^{67} +(3.00000 + 5.19615i) q^{69} +(-9.00000 + 5.19615i) q^{71} +5.19615i q^{73} +(1.00000 - 1.73205i) q^{75} +12.0000 q^{77} -8.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} -3.46410i q^{83} +(-4.50000 + 2.59808i) q^{85} +(4.50000 + 7.79423i) q^{87} +(6.00000 + 3.46410i) q^{89} +(-3.00000 - 5.19615i) q^{95} +(-6.00000 + 3.46410i) q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 6 q^{7} - q^{9}+O(q^{10}) \) 2 * q + q^3 - 6 * q^7 - q^9	\( 2 q + q^{3} - 6 q^{7} - q^{9} - 6 q^{11} - 3 q^{15} - 3 q^{17} + 6 q^{19} - 6 q^{23} + 4 q^{25} - 2 q^{27} - 9 q^{29} - 6 q^{33} + 6 q^{35} + 9 q^{37} + 15 q^{41} - 2 q^{43} - 3 q^{45} + 5 q^{49} - 6 q^{51} - 18 q^{53} - 6 q^{55} - 24 q^{59} + 11 q^{61} + 6 q^{63} - 18 q^{67} + 6 q^{69} - 18 q^{71} + 2 q^{75} + 24 q^{77} - 16 q^{79} - q^{81} - 9 q^{85} + 9 q^{87} + 12 q^{89} - 6 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + q^3 - 6 * q^7 - q^9 - 6 * q^11 - 3 * q^15 - 3 * q^17 + 6 * q^19 - 6 * q^23 + 4 * q^25 - 2 * q^27 - 9 * q^29 - 6 * q^33 + 6 * q^35 + 9 * q^37 + 15 * q^41 - 2 * q^43 - 3 * q^45 + 5 * q^49 - 6 * q^51 - 18 * q^53 - 6 * q^55 - 24 * q^59 + 11 * q^61 + 6 * q^63 - 18 * q^67 + 6 * q^69 - 18 * q^71 + 2 * q^75 + 24 * q^77 - 16 * q^79 - q^81 - 9 * q^85 + 9 * q^87 + 12 * q^89 - 6 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more