LMFDB - Embedded newform 3564.2.i.g.1189.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3564,2,Mod(1189,3564)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3564, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3564.1189"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3564 = 2^{2} \cdot 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3564.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,-2,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$28.4586832804$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1189.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3564.1189
Dual form			3564.2.i.g.2377.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{11} +(-3.00000 - 5.19615i) q^{13} +4.00000 q^{17} -2.00000 q^{19} +(-4.00000 - 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} -4.00000 q^{35} -6.00000 q^{37} +(-5.00000 + 8.66025i) q^{43} +(1.50000 + 2.59808i) q^{49} -14.0000 q^{53} -2.00000 q^{55} +(-6.00000 - 10.3923i) q^{59} +(7.00000 - 12.1244i) q^{61} +(6.00000 - 10.3923i) q^{65} +(-2.00000 - 3.46410i) q^{67} +6.00000 q^{73} +(-1.00000 - 1.73205i) q^{77} +(-1.00000 + 1.73205i) q^{79} +(8.00000 - 13.8564i) q^{83} +(4.00000 + 6.92820i) q^{85} +14.0000 q^{89} +12.0000 q^{91} +(-2.00000 - 3.46410i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7} - q^{11} - 6 q^{13} + 8 q^{17} - 4 q^{19} - 8 q^{23} + q^{25} - 8 q^{35} - 12 q^{37} - 10 q^{43} + 3 q^{49} - 28 q^{53} - 4 q^{55} - 12 q^{59} + 14 q^{61} + 12 q^{65} - 4 q^{67}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - q^11 - 6 * q^13 + 8 * q^17 - 4 * q^19 - 8 * q^23 + q^25 - 8 * q^35 - 12 * q^37 - 10 * q^43 + 3 * q^49 - 28 * q^53 - 4 * q^55 - 12 * q^59 + 14 * q^61 + 12 * q^65 - 4 * q^67 + 12 * q^73 - 2 * q^77 - 2 * q^79 + 16 * q^83 + 8 * q^85 + 28 * q^89 + 24 * q^91 - 4 * q^95 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3564\mathbb{Z}\right)^\times$.

$n$	$1541$	$1783$	$2917$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	0.998203	−	0.0599153i	$-0.0190830\pi$
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	−0.550990	+	0.834512i	$0.685750\pi$
$6$		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	$0.290043\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$12$		0			0
$13$	−3.00000	−	5.19615i	−0.832050	−	1.44115i	−0.896410	−	0.443227i	$-0.853834\pi$
$13$	−3.00000	−	5.19615i	−0.832050	−	1.44115i	0.0643593	−	0.997927i	$-0.479500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.00000			0.970143			0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000			0.970143			0.485071	+	0.874475i	$0.338794\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	−0.894795	−	0.446476i	$-0.852679\pi$
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	0.0607377	−	0.998154i	$-0.480655\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	−5.00000	+	8.66025i	−0.762493	+	1.32068i	0.179069	+	0.983836i	$0.442691\pi$
$43$	−5.00000	+	8.66025i	−0.762493	+	1.32068i	−0.941562	+	0.336840i	$0.890642\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−14.0000			−1.92305			−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000			−1.92305			−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	−0.931282	−	0.364299i	$-0.881308\pi$
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	0.150148	−	0.988663i	$-0.452025\pi$
$60$		0			0
$61$	7.00000	−	12.1244i	0.896258	−	1.55236i	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	7.00000	−	12.1244i	0.896258	−	1.55236i	0.832240	−	0.554416i	$-0.187058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	−	10.3923i	0.744208	−	1.28901i
$66$		0			0
$67$	−2.00000	−	3.46410i	−0.244339	−	0.423207i	0.717607	−	0.696449i	$-0.245238\pi$
$67$	−2.00000	−	3.46410i	−0.244339	−	0.423207i	−0.961946	+	0.273241i	$0.911904\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.00000	−	1.73205i	−0.113961	−	0.197386i
$78$		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	$0.202555\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.00000	−	13.8564i	0.878114	−	1.52094i	0.0247060	−	0.999695i	$-0.492135\pi$
$83$	8.00000	−	13.8564i	0.878114	−	1.52094i	0.853408	−	0.521243i	$-0.174532\pi$
$84$		0			0
$85$	4.00000	+	6.92820i	0.433861	+	0.751469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.00000	−	3.46410i	−0.205196	−	0.355409i
$96$		0			0
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	0.912317	+	0.409484i	$0.134291\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.00000	−	6.92820i	0.398015	−	0.689382i	−0.595466	−	0.803380i	$-0.703033\pi$
$101$	4.00000	−	6.92820i	0.398015	−	0.689382i	0.993481	+	0.113998i	$0.0363659\pi$
$102$		0			0
$103$	6.00000	+	10.3923i	0.591198	+	1.02398i	0.994071	+	0.108729i	$0.0346780\pi$
$103$	6.00000	+	10.3923i	0.591198	+	1.02398i	−0.402874	+	0.915255i	$0.631989\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.0000			−1.54678			−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000			−1.54678			−0.773389	+	0.633932i	$0.781440\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.00000	−	8.66025i	−0.470360	−	0.814688i	0.529065	−	0.848581i	$-0.322543\pi$
$113$	−5.00000	−	8.66025i	−0.470360	−	0.814688i	−0.999425	+	0.0338931i	$0.989209\pi$
$114$		0			0
$115$	8.00000	−	13.8564i	0.746004	−	1.29212i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.00000	+	6.92820i	−0.366679	+	0.635107i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	−6.00000			−0.532414			−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000			−0.532414			−0.266207	+	0.963916i	$0.585770\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	0.940072	−	0.340977i	$-0.110758\pi$
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	−0.765331	+	0.643637i	$0.777425\pi$
$132$		0			0
$133$	2.00000	−	3.46410i	0.173422	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.00000	−	12.1244i	0.598050	−	1.03585i	−0.395058	−	0.918656i	$-0.629276\pi$
$137$	7.00000	−	12.1244i	0.598050	−	1.03585i	0.993109	−	0.117198i	$-0.0373911\pi$
$138$		0			0
$139$	−7.00000	−	12.1244i	−0.593732	−	1.02837i	−0.993724	−	0.111856i	$-0.964321\pi$
$139$	−7.00000	−	12.1244i	−0.593732	−	1.02837i	0.399992	−	0.916519i	$-0.369013\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00000			0.501745
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	0.772399	−	0.635138i	$-0.219057\pi$
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	−0.936245	+	0.351348i	$0.885723\pi$
$150$		0			0
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	−0.994540	−	0.104357i	$-0.966722\pi$
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	0.587646	+	0.809118i	$0.300055\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.00000	+	8.66025i	0.399043	+	0.691164i	0.993608	−	0.112884i	$-0.0360089\pi$
$157$	5.00000	+	8.66025i	0.399043	+	0.691164i	−0.594565	+	0.804048i	$0.702676\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.0000			1.26098
$162$		0			0
$163$	16.0000			1.25322			0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000			1.25322			0.626608	+	0.779334i	$0.284443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	0.534875	−	0.844931i	$-0.320359\pi$
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	−0.999169	+	0.0407502i	$0.987025\pi$
$168$		0			0
$169$	−11.5000	+	19.9186i	−0.884615	+	1.53220i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.00000	−	6.92820i	0.304114	−	0.526742i	−0.672949	−	0.739689i	$-0.734973\pi$
$173$	4.00000	−	6.92820i	0.304114	−	0.526742i	0.977064	+	0.212947i	$0.0683062\pi$
$174$		0			0
$175$	1.00000	+	1.73205i	0.0755929	+	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.00000			0.298974			0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000			0.298974			0.149487	+	0.988764i	$0.452238\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.00000	−	10.3923i	−0.441129	−	0.764057i
$186$		0			0
$187$	−2.00000	+	3.46410i	−0.146254	+	0.253320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.0000	−	20.7846i	0.868290	−	1.50392i	0.00454614	−	0.999990i	$-0.498553\pi$
$191$	12.0000	−	20.7846i	0.868290	−	1.50392i	0.863743	−	0.503932i	$-0.168114\pi$
$192$		0			0
$193$	−5.00000	−	8.66025i	−0.359908	−	0.623379i	0.628037	−	0.778183i	$-0.283859\pi$
$193$	−5.00000	−	8.66025i	−0.359908	−	0.623379i	−0.987945	+	0.154805i	$0.950525\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.0000			−1.13995			−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000			−1.13995			−0.569976	+	0.821661i	$0.693048\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.00000	−	1.73205i	0.0691714	−	0.119808i
$210$		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
$215$	−20.0000			−1.36399
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0000	−	20.7846i	−0.807207	−	1.39812i
$222$		0			0
$223$	−6.00000	+	10.3923i	−0.401790	+	0.695920i	−0.993942	−	0.109906i	$-0.964945\pi$
$223$	−6.00000	+	10.3923i	−0.401790	+	0.695920i	0.592152	+	0.805826i	$0.298278\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000	+	20.7846i	−0.796468	+	1.37952i	0.125435	+	0.992102i	$0.459967\pi$
$227$	−12.0000	+	20.7846i	−0.796468	+	1.37952i	−0.921903	+	0.387421i	$0.873366\pi$
$228$		0			0
$229$	−1.00000	−	1.73205i	−0.0660819	−	0.114457i	0.831092	−	0.556136i	$-0.187717\pi$
$229$	−1.00000	−	1.73205i	−0.0660819	−	0.114457i	−0.897173	+	0.441679i	$0.854383\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.0000			−1.04819			−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000			−1.04819			−0.524097	+	0.851658i	$0.675597\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.00000	−	3.46410i	−0.129369	−	0.224074i	0.794063	−	0.607835i	$-0.207962\pi$
$239$	−2.00000	−	3.46410i	−0.129369	−	0.224074i	−0.923432	+	0.383761i	$0.874629\pi$
$240$		0			0
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	−0.832019	−	0.554747i	$-0.812815\pi$
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	0.896435	+	0.443176i	$0.146148\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.00000	+	5.19615i	−0.191663	+	0.331970i
$246$		0			0
$247$	6.00000	+	10.3923i	0.381771	+	0.661247i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$	8.00000			0.502956
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.00000	+	1.73205i	0.0623783	+	0.108042i	0.895528	−	0.445005i	$-0.146798\pi$
$257$	1.00000	+	1.73205i	0.0623783	+	0.108042i	−0.833150	+	0.553047i	$0.813465\pi$
$258$		0			0
$259$	6.00000	−	10.3923i	0.372822	−	0.645746i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	−14.0000	−	24.2487i	−0.860013	−	1.48959i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$	−10.0000			−0.607457			−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000			−0.607457			−0.303728	+	0.952759i	$0.598232\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.500000	+	0.866025i	0.0301511	+	0.0522233i
$276$		0			0
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	−0.675810	−	0.737075i	$-0.736206\pi$
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	0.976231	+	0.216731i	$0.0695395\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0000	−	27.7128i	0.954480	−	1.65321i	0.218926	−	0.975741i	$-0.429745\pi$
$281$	16.0000	−	27.7128i	0.954480	−	1.65321i	0.735554	−	0.677466i	$-0.236922\pi$
$282$		0			0
$283$	7.00000	+	12.1244i	0.416107	+	0.720718i	0.995544	−	0.0942988i	$-0.0300609\pi$
$283$	7.00000	+	12.1244i	0.416107	+	0.720718i	−0.579437	+	0.815017i	$0.696728\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.0000	−	17.3205i	−0.584206	−	1.01187i	−0.994974	−	0.100135i	$-0.968073\pi$
$293$	−10.0000	−	17.3205i	−0.584206	−	1.01187i	0.410768	−	0.911740i	$-0.365261\pi$
$294$		0			0
$295$	12.0000	−	20.7846i	0.698667	−	1.21013i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−24.0000	+	41.5692i	−1.38796	+	2.40401i
$300$		0			0
$301$	−10.0000	−	17.3205i	−0.576390	−	0.998337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	28.0000			1.60328
$306$		0			0
$307$	−26.0000			−1.48390			−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000			−1.48390			−0.741949	+	0.670456i	$0.766098\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.00000	−	6.92820i	−0.226819	−	0.392862i	0.730044	−	0.683400i	$-0.239499\pi$
$311$	−4.00000	−	6.92820i	−0.226819	−	0.392862i	−0.956864	+	0.290537i	$0.906166\pi$
$312$		0			0
$313$	−7.00000	+	12.1244i	−0.395663	+	0.685309i	−0.993186	−	0.116543i	$-0.962819\pi$
$313$	−7.00000	+	12.1244i	−0.395663	+	0.685309i	0.597522	+	0.801852i	$0.296152\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	−0.494489	−	0.869184i	$-0.664645\pi$
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	0.999980	−	0.00635137i	$-0.00202172\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.00000			−0.445132
$324$		0			0
$325$	−6.00000			−0.332820
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	−0.954765	−	0.297361i	$-0.903893\pi$
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	0.734905	+	0.678170i	$0.237227\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.00000	−	6.92820i	0.218543	−	0.378528i
$336$		0			0
$337$	9.00000	+	15.5885i	0.490261	+	0.849157i	0.999937	−	0.0112091i	$-0.00356804\pi$
$337$	9.00000	+	15.5885i	0.490261	+	0.849157i	−0.509676	+	0.860366i	$0.670235\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	16.0000	+	27.7128i	0.858925	+	1.48770i	0.872955	+	0.487800i	$0.162201\pi$
$347$	16.0000	+	27.7128i	0.858925	+	1.48770i	−0.0140303	+	0.999902i	$0.504466\pi$
$348$		0			0
$349$	3.00000	−	5.19615i	0.160586	−	0.278144i	−0.774493	−	0.632583i	$-0.781995\pi$
$349$	3.00000	−	5.19615i	0.160586	−	0.278144i	0.935079	+	0.354439i	$0.115328\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	−0.934751	−	0.355303i	$-0.884378\pi$
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	0.775077	+	0.631867i	$0.217711\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.00000	+	10.3923i	0.314054	+	0.543958i
$366$		0			0
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	0.913493	+	0.406855i	$0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	14.0000	−	24.2487i	0.726844	−	1.25893i
$372$		0			0
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	0.707055	−	0.707159i	$-0.250023\pi$
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	−0.965945	+	0.258748i	$0.916690\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.00000	+	6.92820i	0.204390	+	0.354015i	0.949938	−	0.312437i	$-0.101145\pi$
$383$	4.00000	+	6.92820i	0.204390	+	0.354015i	−0.745548	+	0.666452i	$0.767812\pi$
$384$		0			0
$385$	2.00000	−	3.46410i	0.101929	−	0.176547i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	−0.542445	−	0.840091i	$-0.682501\pi$
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	0.998763	+	0.0497253i	$0.0158346\pi$
$390$		0			0
$391$	−16.0000	−	27.7128i	−0.809155	−	1.40150i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.00000			−0.201262
$396$		0			0
$397$	−14.0000			−0.702640			−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000			−0.702640			−0.351320	+	0.936255i	$0.614267\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.00000	−	1.73205i	−0.0499376	−	0.0864945i	0.839976	−	0.542623i	$-0.182569\pi$
$401$	−1.00000	−	1.73205i	−0.0499376	−	0.0864945i	−0.889914	+	0.456129i	$0.849236\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.00000	−	5.19615i	0.148704	−	0.257564i
$408$		0			0
$409$	−11.0000	−	19.0526i	−0.543915	−	0.942088i	−0.998674	−	0.0514740i	$-0.983608\pi$
$409$	−11.0000	−	19.0526i	−0.543915	−	0.942088i	0.454759	−	0.890614i	$-0.349725\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	24.0000			1.18096
$414$		0			0
$415$	32.0000			1.57082
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	0.974546	−	0.224189i	$-0.0719734\pi$
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	−0.681426	+	0.731887i	$0.738640\pi$
$420$		0			0
$421$	5.00000	−	8.66025i	0.243685	−	0.422075i	−0.718076	−	0.695965i	$-0.754977\pi$
$421$	5.00000	−	8.66025i	0.243685	−	0.422075i	0.961761	+	0.273890i	$0.0883103\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.00000	−	3.46410i	0.0970143	−	0.168034i
$426$		0			0
$427$	14.0000	+	24.2487i	0.677507	+	1.17348i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.00000	+	13.8564i	0.382692	+	0.662842i
$438$		0			0
$439$	1.00000	−	1.73205i	0.0477274	−	0.0826663i	−0.841175	−	0.540763i	$-0.818135\pi$
$439$	1.00000	−	1.73205i	0.0477274	−	0.0826663i	0.888902	+	0.458097i	$0.151469\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.00000	−	3.46410i	0.0950229	−	0.164584i	−0.814595	−	0.580030i	$-0.803041\pi$
$443$	2.00000	−	3.46410i	0.0950229	−	0.164584i	0.909618	+	0.415445i	$0.136374\pi$
$444$		0			0
$445$	14.0000	+	24.2487i	0.663664	+	1.14950i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	22.0000			1.03824			0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000			1.03824			0.519122	+	0.854700i	$0.326259\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.0000	+	20.7846i	0.562569	+	0.974398i
$456$		0			0
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	0.485299	+	0.874348i	$0.338711\pi$
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	−0.999857	+	0.0168929i	$0.994623\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.0000	+	24.2487i	−0.652045	+	1.12938i	0.330581	+	0.943778i	$0.392755\pi$
$461$	−14.0000	+	24.2487i	−0.652045	+	1.12938i	−0.982626	+	0.185597i	$0.940578\pi$
$462$		0			0
$463$	6.00000	+	10.3923i	0.278844	+	0.482971i	0.971098	−	0.238683i	$-0.0767155\pi$
$463$	6.00000	+	10.3923i	0.278844	+	0.482971i	−0.692254	+	0.721654i	$0.743382\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.00000	−	8.66025i	−0.229900	−	0.398199i
$474$		0			0
$475$	−1.00000	+	1.73205i	−0.0458831	+	0.0794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	−0.969920	−	0.243426i	$-0.921729\pi$
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	0.695773	+	0.718262i	$0.255062\pi$
$480$		0			0
$481$	18.0000	+	31.1769i	0.820729	+	1.42154i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000			0.181631
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.00000	−	13.8564i	−0.361035	−	0.625331i	0.627096	−	0.778942i	$-0.284243\pi$
$491$	−8.00000	−	13.8564i	−0.361035	−	0.625331i	−0.988131	+	0.153611i	$0.950910\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	0.907314	−	0.420455i	$-0.138129\pi$
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	−0.817781	+	0.575529i	$0.804796\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−20.0000			−0.891756			−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000			−0.891756			−0.445878	+	0.895094i	$0.647108\pi$
$504$		0			0
$505$	16.0000			0.711991
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.00000	+	12.1244i	0.310270	+	0.537403i	0.978421	−	0.206623i	$-0.0662474\pi$
$509$	7.00000	+	12.1244i	0.310270	+	0.537403i	−0.668151	+	0.744026i	$0.732914\pi$
$510$		0			0
$511$	−6.00000	+	10.3923i	−0.265424	+	0.459728i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.0000	+	20.7846i	−0.528783	+	0.915879i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$		0			0
$523$	−22.0000			−0.961993			−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000			−0.961993			−0.480996	+	0.876723i	$0.659725\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−20.5000	+	35.5070i	−0.891304	+	1.54378i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−16.0000	−	27.7128i	−0.691740	−	1.19813i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.00000			−0.129219
$540$		0			0
$541$	−26.0000			−1.11783			−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000			−1.11783			−0.558914	+	0.829226i	$0.688782\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−18.0000	−	31.1769i	−0.771035	−	1.33547i
$546$		0			0
$547$	11.0000	−	19.0526i	0.470326	−	0.814629i	−0.529098	−	0.848561i	$-0.677470\pi$
$547$	11.0000	−	19.0526i	0.470326	−	0.814629i	0.999424	+	0.0339321i	$0.0108030\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.00000	−	3.46410i	−0.0850487	−	0.147309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.0000			−0.508456			−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000			−0.508456			−0.254228	+	0.967144i	$0.581821\pi$
$558$		0			0
$559$	60.0000			2.53773
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	0.905088	−	0.425223i	$-0.139804\pi$
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	−0.820798	+	0.571218i	$0.806471\pi$
$564$		0			0
$565$	10.0000	−	17.3205i	0.420703	−	0.728679i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.0000	+	17.3205i	−0.419222	+	0.726113i	−0.995861	−	0.0908852i	$-0.971030\pi$
$569$	−10.0000	+	17.3205i	−0.419222	+	0.726113i	0.576640	+	0.816999i	$0.304364\pi$
$570$		0			0
$571$	−17.0000	−	29.4449i	−0.711428	−	1.23223i	−0.964321	−	0.264735i	$-0.914716\pi$
$571$	−17.0000	−	29.4449i	−0.711428	−	1.23223i	0.252893	−	0.967494i	$-0.418618\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.00000			−0.333623
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.0000	+	27.7128i	0.663792	+	1.14972i
$582$		0			0
$583$	7.00000	−	12.1244i	0.289910	−	0.502140i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.00000	+	10.3923i	−0.247647	+	0.428936i	−0.962872	−	0.269957i	$-0.912990\pi$
$587$	−6.00000	+	10.3923i	−0.247647	+	0.428936i	0.715226	+	0.698893i	$0.246324\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.0000			−0.985562			−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000			−0.985562			−0.492781	+	0.870153i	$0.664020\pi$
$594$		0			0
$595$	−16.0000			−0.655936
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	7.00000	−	12.1244i	0.285536	−	0.494563i	−0.687203	−	0.726465i	$-0.741162\pi$
$601$	7.00000	−	12.1244i	0.285536	−	0.494563i	0.972739	+	0.231903i	$0.0744951\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.00000	−	1.73205i	0.0406558	−	0.0704179i
$606$		0			0
$607$	−17.0000	−	29.4449i	−0.690009	−	1.19513i	−0.971834	−	0.235665i	$-0.924273\pi$
$607$	−17.0000	−	29.4449i	−0.690009	−	1.19513i	0.281826	−	0.959466i	$-0.409060\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	42.0000			1.69636			0.848182	−	0.529705i	$-0.177697\pi$
$613$	42.0000			1.69636			0.848182	+	0.529705i	$0.177697\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	19.0000	+	32.9090i	0.764911	+	1.32487i	0.940294	+	0.340365i	$0.110551\pi$
$617$	19.0000	+	32.9090i	0.764911	+	1.32487i	−0.175382	+	0.984500i	$0.556116\pi$
$618$		0			0
$619$	−16.0000	+	27.7128i	−0.643094	+	1.11387i	0.341644	+	0.939829i	$0.389016\pi$
$619$	−16.0000	+	27.7128i	−0.643094	+	1.11387i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.0000	+	24.2487i	−0.560898	+	0.971504i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24.0000			−0.956943
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.00000	−	10.3923i	−0.238103	−	0.412406i
$636$		0			0
$637$	9.00000	−	15.5885i	0.356593	−	0.617637i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.00000	+	5.19615i	−0.118493	+	0.205236i	−0.919171	−	0.393860i	$-0.871140\pi$
$641$	−3.00000	+	5.19615i	−0.118493	+	0.205236i	0.800678	+	0.599095i	$0.204473\pi$
$642$		0			0
$643$	−16.0000	−	27.7128i	−0.630978	−	1.09289i	−0.987352	−	0.158543i	$-0.949320\pi$
$643$	−16.0000	−	27.7128i	−0.630978	−	1.09289i	0.356374	−	0.934344i	$-0.384013\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.00000			−0.314512			−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000			−0.314512			−0.157256	+	0.987558i	$0.550265\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21.0000	−	36.3731i	−0.821794	−	1.42339i	−0.904345	−	0.426801i	$-0.859640\pi$
$653$	−21.0000	−	36.3731i	−0.821794	−	1.42339i	0.0825519	−	0.996587i	$-0.473693\pi$
$654$		0			0
$655$	−4.00000	+	6.92820i	−0.156293	+	0.270707i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	−0.958902	−	0.283738i	$-0.908425\pi$
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	0.725175	+	0.688565i	$0.241759\pi$
$660$		0			0
$661$	9.00000	+	15.5885i	0.350059	+	0.606321i	0.986260	−	0.165203i	$-0.0528281\pi$
$661$	9.00000	+	15.5885i	0.350059	+	0.606321i	−0.636200	+	0.771524i	$0.719495\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.00000			0.310227
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.00000	+	12.1244i	0.270232	+	0.468056i
$672$		0			0
$673$	−13.0000	+	22.5167i	−0.501113	+	0.867953i	0.498886	+	0.866668i	$0.333743\pi$
$673$	−13.0000	+	22.5167i	−0.501113	+	0.867953i	−0.999999	+	0.00128586i	$0.999591\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.0000	+	27.7128i	−0.614930	+	1.06509i	0.375467	+	0.926836i	$0.377482\pi$
$677$	−16.0000	+	27.7128i	−0.614930	+	1.06509i	−0.990397	+	0.138254i	$0.955851\pi$
$678$		0			0
$679$	2.00000	+	3.46410i	0.0767530	+	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$	28.0000			1.06983
$686$		0			0
$687$		0			0
$688$		0			0
$689$	42.0000	+	72.7461i	1.60007	+	2.77141i
$690$		0			0
$691$	−12.0000	+	20.7846i	−0.456502	+	0.790684i	−0.998773	−	0.0495194i	$-0.984231\pi$
$691$	−12.0000	+	20.7846i	−0.456502	+	0.790684i	0.542272	+	0.840203i	$0.317564\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14.0000	−	24.2487i	0.531050	−	0.919806i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$	12.0000			0.452589
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.00000	+	13.8564i	0.300871	+	0.521124i
$708$		0			0
$709$	−9.00000	+	15.5885i	−0.338002	+	0.585437i	−0.984057	−	0.177854i	$-0.943084\pi$
$709$	−9.00000	+	15.5885i	−0.338002	+	0.585437i	0.646055	+	0.763291i	$0.276418\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	6.00000	+	10.3923i	0.224387	+	0.388650i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	24.0000	−	41.5692i	0.890111	−	1.54172i	0.0503692	−	0.998731i	$-0.483960\pi$
$727$	24.0000	−	41.5692i	0.890111	−	1.54172i	0.839742	−	0.542986i	$-0.182706\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−20.0000	+	34.6410i	−0.739727	+	1.28124i
$732$		0			0
$733$	17.0000	+	29.4449i	0.627909	+	1.08757i	0.987971	+	0.154642i	$0.0494225\pi$
$733$	17.0000	+	29.4449i	0.627909	+	1.08757i	−0.360061	+	0.932929i	$0.617244\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.00000			0.147342
$738$		0			0
$739$	26.0000			0.956425			0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000			0.956425			0.478213	+	0.878244i	$0.341285\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.00000	−	6.92820i	−0.146746	−	0.254171i	0.783277	−	0.621673i	$-0.213547\pi$
$743$	−4.00000	−	6.92820i	−0.146746	−	0.254171i	−0.930023	+	0.367502i	$0.880213\pi$
$744$		0			0
$745$	4.00000	−	6.92820i	0.146549	−	0.253830i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	16.0000	−	27.7128i	0.584627	−	1.01260i
$750$		0			0
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	0.956963	+	0.290209i	$0.0937250\pi$
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	−0.227153	+	0.973859i	$0.572942\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−20.0000			−0.727875
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.0000	+	31.1769i	0.652499	+	1.13016i	0.982514	+	0.186187i	$0.0596129\pi$
$761$	18.0000	+	31.1769i	0.652499	+	1.13016i	−0.330015	+	0.943976i	$0.607054\pi$
$762$		0			0
$763$	18.0000	−	31.1769i	0.651644	−	1.12868i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−36.0000	+	62.3538i	−1.29988	+	2.25147i
$768$		0			0
$769$	−17.0000	−	29.4449i	−0.613036	−	1.06181i	−0.990726	−	0.135877i	$-0.956615\pi$
$769$	−17.0000	−	29.4449i	−0.613036	−	1.06181i	0.377690	−	0.925932i	$-0.376718\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.0000	+	17.3205i	−0.356915	+	0.618195i
$786$		0			0
$787$	19.0000	+	32.9090i	0.677277	+	1.17308i	0.975798	+	0.218675i	$0.0701734\pi$
$787$	19.0000	+	32.9090i	0.677277	+	1.17308i	−0.298521	+	0.954403i	$0.596493\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20.0000			0.711118
$792$		0			0
$793$	−84.0000			−2.98293
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.00000	−	5.19615i	−0.106265	−	0.184057i	0.807989	−	0.589197i	$-0.200556\pi$
$797$	−3.00000	−	5.19615i	−0.106265	−	0.184057i	−0.914255	+	0.405140i	$0.867223\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.00000	+	5.19615i	−0.105868	+	0.183368i
$804$		0			0
$805$	16.0000	+	27.7128i	0.563926	+	0.976748i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	44.0000			1.54696			0.773479	−	0.633822i	$-0.218515\pi$
$809$	44.0000			1.54696			0.773479	+	0.633822i	$0.218515\pi$
$810$		0			0
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.0000	+	27.7128i	0.560456	+	0.970737i
$816$		0			0
$817$	10.0000	−	17.3205i	0.349856	−	0.605968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	20.0000	−	34.6410i	0.698005	−	1.20898i	−0.271152	−	0.962536i	$-0.587405\pi$
$821$	20.0000	−	34.6410i	0.698005	−	1.20898i	0.969157	−	0.246443i	$-0.0792619\pi$
$822$		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	$0.671057\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.0000			0.556375			0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000			0.556375			0.278187	+	0.960527i	$0.410266\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.00000	+	10.3923i	0.207888	+	0.360072i
$834$		0			0
$835$	12.0000	−	20.7846i	0.415277	−	0.719281i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.0000	+	20.7846i	−0.414286	+	0.717564i	−0.995353	−	0.0962912i	$-0.969302\pi$
$839$	−12.0000	+	20.7846i	−0.414286	+	0.717564i	0.581067	+	0.813856i	$0.302635\pi$
$840$		0			0
$841$	14.5000	+	25.1147i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−46.0000			−1.58245
$846$		0			0
$847$	2.00000			0.0687208
$848$		0			0
$849$		0			0
$850$		0			0
$851$	24.0000	+	41.5692i	0.822709	+	1.42497i
$852$		0			0
$853$	−9.00000	+	15.5885i	−0.308154	+	0.533739i	−0.977959	−	0.208799i	$-0.933045\pi$
$853$	−9.00000	+	15.5885i	−0.308154	+	0.533739i	0.669804	+	0.742538i	$0.266378\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.0000	−	38.1051i	0.751506	−	1.30165i	−0.195587	−	0.980686i	$-0.562661\pi$
$857$	22.0000	−	38.1051i	0.751506	−	1.30165i	0.947093	−	0.320960i	$-0.104005\pi$
$858$		0			0
$859$	−20.0000	−	34.6410i	−0.682391	−	1.18194i	−0.974249	−	0.225475i	$-0.927607\pi$
$859$	−20.0000	−	34.6410i	−0.682391	−	1.18194i	0.291858	−	0.956462i	$-0.405727\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−8.00000			−0.272323			−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000			−0.272323			−0.136162	+	0.990687i	$0.543477\pi$
$864$		0			0
$865$	16.0000			0.544016
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.00000	−	1.73205i	−0.0339227	−	0.0587558i
$870$		0			0
$871$	−12.0000	+	20.7846i	−0.406604	+	0.704260i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.0000	+	20.7846i	−0.405674	+	0.702648i
$876$		0			0
$877$	−9.00000	−	15.5885i	−0.303908	−	0.526385i	0.673109	−	0.739543i	$-0.264958\pi$
$877$	−9.00000	−	15.5885i	−0.303908	−	0.526385i	−0.977018	+	0.213158i	$0.931625\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		0			0
$883$	−44.0000			−1.48072			−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000			−1.48072			−0.740359	+	0.672212i	$0.765344\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	6.00000	−	10.3923i	0.201234	−	0.348547i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	4.00000	+	6.92820i	0.133705	+	0.231584i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−56.0000			−1.86563
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.0000	+	38.1051i	0.731305	+	1.26666i
$906$		0			0
$907$	−8.00000	+	13.8564i	−0.265636	+	0.460094i	−0.967730	−	0.251990i	$-0.918915\pi$
$907$	−8.00000	+	13.8564i	−0.265636	+	0.460094i	0.702094	+	0.712084i	$0.252248\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.0000	+	41.5692i	−0.795155	+	1.37725i	0.127585	+	0.991828i	$0.459277\pi$
$911$	−24.0000	+	41.5692i	−0.795155	+	1.37725i	−0.922740	+	0.385422i	$0.874056\pi$
$912$		0			0
$913$	8.00000	+	13.8564i	0.264761	+	0.458580i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.00000			−0.264183
$918$		0			0
$919$	−34.0000			−1.12156			−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000			−1.12156			−0.560778	+	0.827966i	$0.689498\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−3.00000	+	5.19615i	−0.0986394	+	0.170848i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−21.0000	+	36.3731i	−0.688988	+	1.19336i	0.283178	+	0.959067i	$0.408611\pi$
$929$	−21.0000	+	36.3731i	−0.688988	+	1.19336i	−0.972166	+	0.234294i	$0.924722\pi$
$930$		0			0
$931$	−3.00000	−	5.19615i	−0.0983210	−	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.00000			−0.261628
$936$		0			0
$937$	−38.0000			−1.24141			−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000			−1.24141			−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.00000	−	10.3923i	0.194974	−	0.337705i	−0.751918	−	0.659256i	$-0.770871\pi$
$947$	6.00000	−	10.3923i	0.194974	−	0.337705i	0.946892	+	0.321552i	$0.104204\pi$
$948$		0			0
$949$	−18.0000	−	31.1769i	−0.584305	−	1.01205i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−12.0000			−0.388718			−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000			−0.388718			−0.194359	+	0.980930i	$0.562263\pi$
$954$		0			0
$955$	48.0000			1.55324
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.0000	+	24.2487i	0.452084	+	0.783032i
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.0000	−	17.3205i	0.321911	−	0.557567i
$966$		0			0
$967$	−17.0000	−	29.4449i	−0.546683	−	0.946883i	−0.998499	−	0.0547717i	$-0.982557\pi$
$967$	−17.0000	−	29.4449i	−0.546683	−	0.946883i	0.451816	−	0.892111i	$-0.350776\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−60.0000			−1.92549			−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000			−1.92549			−0.962746	+	0.270408i	$0.912841\pi$
$972$		0			0
$973$	28.0000			0.897639
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.00000	−	8.66025i	−0.159964	−	0.277066i	0.774891	−	0.632094i	$-0.217805\pi$
$977$	−5.00000	−	8.66025i	−0.159964	−	0.277066i	−0.934856	+	0.355028i	$0.884471\pi$
$978$		0			0
$979$	−7.00000	+	12.1244i	−0.223721	+	0.387496i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	−16.0000	−	27.7128i	−0.509802	−	0.883004i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	80.0000			2.54385
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.00000	+	13.8564i	0.253617	+	0.439278i
$996$		0			0
$997$	−5.00000	+	8.66025i	−0.158352	+	0.274273i	−0.934274	−	0.356555i	$-0.883951\pi$
$997$	−5.00000	+	8.66025i	−0.158352	+	0.274273i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3564.2.i.g.1189.1		2
3.2	odd	2		3564.2.i.b.1189.1		2
9.2	odd	6		132.2.a.a.1.1	✓	1
9.4	even	3	inner	3564.2.i.g.2377.1		2
9.5	odd	6		3564.2.i.b.2377.1		2
9.7	even	3		396.2.a.b.1.1		1
36.7	odd	6		1584.2.a.c.1.1		1
36.11	even	6		528.2.a.h.1.1		1
45.2	even	12		3300.2.c.c.1849.2		2
45.7	odd	12		9900.2.c.k.5149.2		2
45.29	odd	6		3300.2.a.k.1.1		1
45.34	even	6		9900.2.a.g.1.1		1
45.38	even	12		3300.2.c.c.1849.1		2
45.43	odd	12		9900.2.c.k.5149.1		2
63.20	even	6		6468.2.a.l.1.1		1
72.11	even	6		2112.2.a.b.1.1		1
72.29	odd	6		2112.2.a.t.1.1		1
72.43	odd	6		6336.2.a.bz.1.1		1
72.61	even	6		6336.2.a.cf.1.1		1
99.2	even	30		1452.2.i.l.565.1		4
99.20	odd	30		1452.2.i.k.565.1		4
99.29	even	30		1452.2.i.l.1237.1		4
99.38	odd	30		1452.2.i.k.1213.1		4
99.43	odd	6		4356.2.a.c.1.1		1
99.47	odd	30		1452.2.i.k.493.1		4
99.65	even	6		1452.2.a.c.1.1		1
99.74	even	30		1452.2.i.l.493.1		4
99.83	even	30		1452.2.i.l.1213.1		4
99.92	odd	30		1452.2.i.k.1237.1		4
396.263	odd	6		5808.2.a.bf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.2.a.a.1.1	✓	1	9.2	odd	6
396.2.a.b.1.1		1	9.7	even	3
528.2.a.h.1.1		1	36.11	even	6
1452.2.a.c.1.1		1	99.65	even	6
1452.2.i.k.493.1		4	99.47	odd	30
1452.2.i.k.565.1		4	99.20	odd	30
1452.2.i.k.1213.1		4	99.38	odd	30
1452.2.i.k.1237.1		4	99.92	odd	30
1452.2.i.l.493.1		4	99.74	even	30
1452.2.i.l.565.1		4	99.2	even	30
1452.2.i.l.1213.1		4	99.83	even	30
1452.2.i.l.1237.1		4	99.29	even	30
1584.2.a.c.1.1		1	36.7	odd	6
2112.2.a.b.1.1		1	72.11	even	6
2112.2.a.t.1.1		1	72.29	odd	6
3300.2.a.k.1.1		1	45.29	odd	6
3300.2.c.c.1849.1		2	45.38	even	12
3300.2.c.c.1849.2		2	45.2	even	12
3564.2.i.b.1189.1		2	3.2	odd	2
3564.2.i.b.2377.1		2	9.5	odd	6
3564.2.i.g.1189.1		2	1.1	even	1	trivial
3564.2.i.g.2377.1		2	9.4	even	3	inner
4356.2.a.c.1.1		1	99.43	odd	6
5808.2.a.bf.1.1		1	396.263	odd	6
6336.2.a.bz.1.1		1	72.43	odd	6
6336.2.a.cf.1.1		1	72.61	even	6
6468.2.a.l.1.1		1	63.20	even	6
9900.2.a.g.1.1		1	45.34	even	6
9900.2.c.k.5149.1		2	45.43	odd	12
9900.2.c.k.5149.2		2	45.7	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3564 = 2^{2} \cdot 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3564.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{11} +(-3.00000 - 5.19615i) q^{13} +4.00000 q^{17} -2.00000 q^{19} +(-4.00000 - 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} -4.00000 q^{35} -6.00000 q^{37} +(-5.00000 + 8.66025i) q^{43} +(1.50000 + 2.59808i) q^{49} -14.0000 q^{53} -2.00000 q^{55} +(-6.00000 - 10.3923i) q^{59} +(7.00000 - 12.1244i) q^{61} +(6.00000 - 10.3923i) q^{65} +(-2.00000 - 3.46410i) q^{67} +6.00000 q^{73} +(-1.00000 - 1.73205i) q^{77} +(-1.00000 + 1.73205i) q^{79} +(8.00000 - 13.8564i) q^{83} +(4.00000 + 6.92820i) q^{85} +14.0000 q^{89} +12.0000 q^{91} +(-2.00000 - 3.46410i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} - q^{11} - 6 q^{13} + 8 q^{17} - 4 q^{19} - 8 q^{23} + q^{25} - 8 q^{35} - 12 q^{37} - 10 q^{43} + 3 q^{49} - 28 q^{53} - 4 q^{55} - 12 q^{59} + 14 q^{61} + 12 q^{65} - 4 q^{67}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - q^11 - 6 * q^13 + 8 * q^17 - 4 * q^19 - 8 * q^23 + q^25 - 8 * q^35 - 12 * q^37 - 10 * q^43 + 3 * q^49 - 28 * q^53 - 4 * q^55 - 12 * q^59 + 14 * q^61 + 12 * q^65 - 4 * q^67 + 12 * q^73 - 2 * q^77 - 2 * q^79 + 16 * q^83 + 8 * q^85 + 28 * q^89 + 24 * q^91 - 4 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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