LMFDB - Embedded newform 1568.2.q.e.815.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,2,Mod(815,1568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1568.815"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,12,0,8,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			815.3
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1568.815
Dual form			1568.2.q.e.1391.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.12132 + 1.22474i) q^{3} +(-1.22474 - 2.12132i) q^{5} +(1.50000 + 2.59808i) q^{9} +(1.00000 - 1.73205i) q^{11} -2.44949 q^{13} -6.00000i q^{15} +(4.24264 + 2.44949i) q^{17} +(2.12132 - 1.22474i) q^{19} +(3.46410 - 2.00000i) q^{23} +(-0.500000 + 0.866025i) q^{25} +4.00000i q^{29} +(2.44949 - 4.24264i) q^{31} +(4.24264 - 2.44949i) q^{33} +(6.92820 - 4.00000i) q^{37} +(-5.19615 - 3.00000i) q^{39} +6.00000 q^{43} +(3.67423 - 6.36396i) q^{45} +(-2.44949 - 4.24264i) q^{47} +(6.00000 + 10.3923i) q^{51} +(3.46410 + 2.00000i) q^{53} -4.89898 q^{55} +6.00000 q^{57} +(-2.12132 - 1.22474i) q^{59} +(3.67423 + 6.36396i) q^{61} +(3.00000 + 5.19615i) q^{65} +(-1.00000 + 1.73205i) q^{67} +9.79796 q^{69} -10.0000i q^{71} +(-12.7279 - 7.34847i) q^{73} +(-2.12132 + 1.22474i) q^{75} +(5.19615 - 3.00000i) q^{79} +(4.50000 - 7.79423i) q^{81} -2.44949i q^{83} -12.0000i q^{85} +(-4.89898 + 8.48528i) q^{87} +(-12.7279 + 7.34847i) q^{89} +(10.3923 - 6.00000i) q^{93} +(-5.19615 - 3.00000i) q^{95} -4.89898i q^{97} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{9} + 8 q^{11} - 4 q^{25} + 48 q^{43} + 48 q^{51} + 48 q^{57} + 24 q^{65} - 8 q^{67} + 36 q^{81} + 48 q^{99}+O(q^{100}) $ 8 * q + 12 * q^9 + 8 * q^11 - 4 * q^25 + 48 * q^43 + 48 * q^51 + 48 * q^57 + 24 * q^65 - 8 * q^67 + 36 * q^81 + 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$3$	2.12132	+	1.22474i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$3$	2.12132	+	1.22474i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$4$		0			0
$5$	−1.22474	−	2.12132i	−0.547723	−	0.948683i	−0.998430	−	0.0560116i	$-0.982162\pi$
$5$	−1.22474	−	2.12132i	−0.547723	−	0.948683i	0.450708	−	0.892672i	$-0.351172\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		0			0
$13$	−2.44949			−0.679366			−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949			−0.679366			−0.339683	+	0.940540i	$0.610320\pi$
$14$		0			0
$15$		−	6.00000i		−	1.54919i
$16$		0			0
$17$	4.24264	+	2.44949i	1.02899	+	0.594089i	0.916696	−	0.399586i	$-0.130846\pi$
$17$	4.24264	+	2.44949i	1.02899	+	0.594089i	0.112296	+	0.993675i	$0.464180\pi$
$18$		0			0
$19$	2.12132	−	1.22474i	0.486664	−	0.280976i	−0.236525	−	0.971625i	$-0.576009\pi$
$19$	2.12132	−	1.22474i	0.486664	−	0.280976i	0.723190	+	0.690650i	$0.242675\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.46410	−	2.00000i	0.722315	−	0.417029i	−0.0932891	−	0.995639i	$-0.529738\pi$
$23$	3.46410	−	2.00000i	0.722315	−	0.417029i	0.815604	+	0.578610i	$0.196405\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$	2.44949	−	4.24264i	0.439941	−	0.762001i	−0.557743	−	0.830014i	$-0.688333\pi$
$31$	2.44949	−	4.24264i	0.439941	−	0.762001i	0.997684	+	0.0680129i	$0.0216659\pi$
$32$		0			0
$33$	4.24264	−	2.44949i	0.738549	−	0.426401i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.92820	−	4.00000i	1.13899	−	0.657596i	0.192809	−	0.981236i	$-0.438240\pi$
$37$	6.92820	−	4.00000i	1.13899	−	0.657596i	0.946180	+	0.323640i	$0.104907\pi$
$38$		0			0
$39$	−5.19615	−	3.00000i	−0.832050	−	0.480384i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$		0			0
$45$	3.67423	−	6.36396i	0.547723	−	0.948683i
$46$		0			0
$47$	−2.44949	−	4.24264i	−0.357295	−	0.618853i	0.630213	−	0.776422i	$-0.282968\pi$
$47$	−2.44949	−	4.24264i	−0.357295	−	0.618853i	−0.987508	+	0.157569i	$0.949634\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	6.00000	+	10.3923i	0.840168	+	1.45521i
$52$		0			0
$53$	3.46410	+	2.00000i	0.475831	+	0.274721i	0.718677	−	0.695344i	$-0.244748\pi$
$53$	3.46410	+	2.00000i	0.475831	+	0.274721i	−0.242846	+	0.970065i	$0.578081\pi$
$54$		0			0
$55$	−4.89898			−0.660578
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$	−2.12132	−	1.22474i	−0.276172	−	0.159448i	0.355517	−	0.934670i	$-0.384305\pi$
$59$	−2.12132	−	1.22474i	−0.276172	−	0.159448i	−0.631689	+	0.775222i	$0.717638\pi$
$60$		0			0
$61$	3.67423	+	6.36396i	0.470438	+	0.814822i	0.999428	−	0.0338058i	$-0.0107628\pi$
$61$	3.67423	+	6.36396i	0.470438	+	0.814822i	−0.528991	+	0.848628i	$0.677429\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	+	5.19615i	0.372104	+	0.644503i
$66$		0			0
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$		0			0
$69$	9.79796			1.17954
$70$		0			0
$71$		−	10.0000i		−	1.18678i	−0.804914	−	0.593391i	$-0.797789\pi$
$71$		−	10.0000i		−	1.18678i	0.804914	−	0.593391i	$-0.202211\pi$
$72$		0			0
$73$	−12.7279	−	7.34847i	−1.48969	−	0.860073i	−0.489760	−	0.871857i	$-0.662916\pi$
$73$	−12.7279	−	7.34847i	−1.48969	−	0.860073i	−0.999931	+	0.0117840i	$0.996249\pi$
$74$		0			0
$75$	−2.12132	+	1.22474i	−0.244949	+	0.141421i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.19615	−	3.00000i	0.584613	−	0.337526i	−0.178352	−	0.983967i	$-0.557076\pi$
$79$	5.19615	−	3.00000i	0.584613	−	0.337526i	0.762964	+	0.646440i	$0.223743\pi$
$80$		0			0
$81$	4.50000	−	7.79423i	0.500000	−	0.866025i
$82$		0			0
$83$		−	2.44949i		−	0.268866i	−0.990923	−	0.134433i	$-0.957079\pi$
$83$		−	2.44949i		−	0.268866i	0.990923	−	0.134433i	$-0.0429214\pi$
$84$		0			0
$85$		−	12.0000i		−	1.30158i
$86$		0			0
$87$	−4.89898	+	8.48528i	−0.525226	+	0.909718i
$88$		0			0
$89$	−12.7279	+	7.34847i	−1.34916	+	0.778936i	−0.988130	−	0.153619i	$-0.950907\pi$
$89$	−12.7279	+	7.34847i	−1.34916	+	0.778936i	−0.361027	+	0.932555i	$0.617574\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	10.3923	−	6.00000i	1.07763	−	0.622171i
$94$		0			0
$95$	−5.19615	−	3.00000i	−0.533114	−	0.307794i
$96$		0			0
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	−8.57321	+	14.8492i	−0.853067	+	1.47755i	0.0253604	+	0.999678i	$0.491927\pi$
$101$	−8.57321	+	14.8492i	−0.853067	+	1.47755i	−0.878427	+	0.477876i	$0.841407\pi$
$102$		0			0
$103$	4.89898	+	8.48528i	0.482711	+	0.836080i	0.999803	−	0.0198501i	$-0.00631890\pi$
$103$	4.89898	+	8.48528i	0.482711	+	0.836080i	−0.517092	+	0.855930i	$0.672986\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.00000	−	1.73205i	−0.0966736	−	0.167444i	0.813632	−	0.581380i	$-0.197487\pi$
$107$	−1.00000	−	1.73205i	−0.0966736	−	0.167444i	−0.910306	+	0.413936i	$0.864154\pi$
$108$		0			0
$109$	−3.46410	−	2.00000i	−0.331801	−	0.191565i	0.324840	−	0.945769i	$-0.394690\pi$
$109$	−3.46410	−	2.00000i	−0.331801	−	0.191565i	−0.656640	+	0.754204i	$0.728023\pi$
$110$		0			0
$111$	19.5959			1.85996
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	−8.48528	−	4.89898i	−0.791257	−	0.456832i
$116$		0			0
$117$	−3.67423	−	6.36396i	−0.339683	−	0.588348i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.79796			−0.876356
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$		0			0
$129$	12.7279	+	7.34847i	1.12063	+	0.646997i
$130$		0			0
$131$	−10.6066	+	6.12372i	−0.926703	+	0.535032i	−0.885767	−	0.464130i	$-0.846367\pi$
$131$	−10.6066	+	6.12372i	−0.926703	+	0.535032i	−0.0409357	+	0.999162i	$0.513034\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	−0.820141	−	0.572161i	$-0.806105\pi$
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	0.905577	+	0.424182i	$0.139438\pi$
$138$		0			0
$139$			2.44949i			0.207763i	0.994590	+	0.103882i	$0.0331263\pi$
$139$			2.44949i			0.207763i	−0.994590	+	0.103882i	$0.966874\pi$
$140$		0			0
$141$		−	12.0000i		−	1.01058i
$142$		0			0
$143$	−2.44949	+	4.24264i	−0.204837	+	0.354787i
$144$		0			0
$145$	8.48528	−	4.89898i	0.704664	−	0.406838i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.8564	+	8.00000i	−1.13516	+	0.655386i	−0.945228	−	0.326411i	$-0.894160\pi$
$149$	−13.8564	+	8.00000i	−1.13516	+	0.655386i	−0.189933	+	0.981797i	$0.560827\pi$
$150$		0			0
$151$	17.3205	+	10.0000i	1.40952	+	0.813788i	0.995342	−	0.0964061i	$-0.0307348\pi$
$151$	17.3205	+	10.0000i	1.40952	+	0.813788i	0.414181	+	0.910195i	$0.364068\pi$
$152$		0			0
$153$			14.6969i			1.18818i
$154$		0			0
$155$	−12.0000			−0.963863
$156$		0			0
$157$	−3.67423	+	6.36396i	−0.293236	+	0.507899i	−0.974573	−	0.224070i	$-0.928065\pi$
$157$	−3.67423	+	6.36396i	−0.293236	+	0.507899i	0.681337	+	0.731970i	$0.261399\pi$
$158$		0			0
$159$	4.89898	+	8.48528i	0.388514	+	0.672927i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	7.00000	+	12.1244i	0.548282	+	0.949653i	0.998392	+	0.0566798i	$0.0180514\pi$
$163$	7.00000	+	12.1244i	0.548282	+	0.949653i	−0.450110	+	0.892973i	$0.648615\pi$
$164$		0			0
$165$	−10.3923	−	6.00000i	−0.809040	−	0.467099i
$166$		0			0
$167$	−19.5959			−1.51638			−0.758189	−	0.652035i	$-0.773915\pi$
$167$	−19.5959			−1.51638			−0.758189	+	0.652035i	$0.773915\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$	6.36396	+	3.67423i	0.486664	+	0.280976i
$172$		0			0
$173$	1.22474	+	2.12132i	0.0931156	+	0.161281i	0.908821	−	0.417187i	$-0.136984\pi$
$173$	1.22474	+	2.12132i	0.0931156	+	0.161281i	−0.815705	+	0.578468i	$0.803651\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−3.00000	−	5.19615i	−0.225494	−	0.390567i
$178$		0			0
$179$	5.00000	−	8.66025i	0.373718	−	0.647298i	−0.616417	−	0.787420i	$-0.711416\pi$
$179$	5.00000	−	8.66025i	0.373718	−	0.647298i	0.990134	+	0.140122i	$0.0447496\pi$
$180$		0			0
$181$	−7.34847			−0.546207			−0.273104	−	0.961985i	$-0.588050\pi$
$181$	−7.34847			−0.546207			−0.273104	+	0.961985i	$0.588050\pi$
$182$		0			0
$183$			18.0000i			1.33060i
$184$		0			0
$185$	−16.9706	−	9.79796i	−1.24770	−	0.720360i
$186$		0			0
$187$	8.48528	−	4.89898i	0.620505	−	0.358249i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.66025	−	5.00000i	0.626634	−	0.361787i	−0.152813	−	0.988255i	$-0.548833\pi$
$191$	8.66025	−	5.00000i	0.626634	−	0.361787i	0.779447	+	0.626468i	$0.215500\pi$
$192$		0			0
$193$	−12.0000	+	20.7846i	−0.863779	+	1.49611i	0.00447566	+	0.999990i	$0.498575\pi$
$193$	−12.0000	+	20.7846i	−0.863779	+	1.49611i	−0.868255	+	0.496119i	$0.834758\pi$
$194$		0			0
$195$			14.6969i			1.05247i
$196$		0			0
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$		0			0
$201$	−4.24264	+	2.44949i	−0.299253	+	0.172774i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	10.3923	+	6.00000i	0.722315	+	0.417029i
$208$		0			0
$209$		−	4.89898i		−	0.338869i
$210$		0			0
$211$	18.0000			1.23917			0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000			1.23917			0.619586	+	0.784929i	$0.287301\pi$
$212$		0			0
$213$	12.2474	−	21.2132i	0.839181	−	1.45350i
$214$		0			0
$215$	−7.34847	−	12.7279i	−0.501161	−	0.868037i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−18.0000	−	31.1769i	−1.21633	−	2.10674i
$220$		0			0
$221$	−10.3923	−	6.00000i	−0.699062	−	0.403604i
$222$		0			0
$223$	−9.79796			−0.656120			−0.328060	−	0.944657i	$-0.606395\pi$
$223$	−9.79796			−0.656120			−0.328060	+	0.944657i	$0.606395\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$	6.36396	+	3.67423i	0.422391	+	0.243868i	0.696100	−	0.717945i	$-0.254917\pi$
$227$	6.36396	+	3.67423i	0.422391	+	0.243868i	−0.273709	+	0.961813i	$0.588250\pi$
$228$		0			0
$229$	6.12372	+	10.6066i	0.404667	+	0.700904i	0.994283	−	0.106780i	$-0.0340541\pi$
$229$	6.12372	+	10.6066i	0.404667	+	0.700904i	−0.589616	+	0.807684i	$0.700721\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.00000	−	12.1244i	−0.458585	−	0.794293i	0.540301	−	0.841472i	$-0.318310\pi$
$233$	−7.00000	−	12.1244i	−0.458585	−	0.794293i	−0.998886	+	0.0471787i	$0.984977\pi$
$234$		0			0
$235$	−6.00000	+	10.3923i	−0.391397	+	0.677919i
$236$		0			0
$237$	14.6969			0.954669
$238$		0			0
$239$		−	4.00000i		−	0.258738i	−0.991596	−	0.129369i	$-0.958705\pi$
$239$		−	4.00000i		−	0.258738i	0.991596	−	0.129369i	$-0.0412952\pi$
$240$		0			0
$241$	21.2132	+	12.2474i	1.36646	+	0.788928i	0.990474	−	0.137697i	$-0.0439700\pi$
$241$	21.2132	+	12.2474i	1.36646	+	0.788928i	0.375988	+	0.926624i	$0.377303\pi$
$242$		0			0
$243$	19.0919	−	11.0227i	1.22474	−	0.707107i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5.19615	+	3.00000i	−0.330623	+	0.190885i
$248$		0			0
$249$	3.00000	−	5.19615i	0.190117	−	0.329293i
$250$		0			0
$251$		−	12.2474i		−	0.773052i	−0.922278	−	0.386526i	$-0.873675\pi$
$251$		−	12.2474i		−	0.773052i	0.922278	−	0.386526i	$-0.126325\pi$
$252$		0			0
$253$		−	8.00000i		−	0.502956i
$254$		0			0
$255$	14.6969	−	25.4558i	0.920358	−	1.59411i
$256$		0			0
$257$	16.9706	−	9.79796i	1.05859	−	0.611180i	0.133551	−	0.991042i	$-0.457362\pi$
$257$	16.9706	−	9.79796i	1.05859	−	0.611180i	0.925043	+	0.379862i	$0.124029\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−10.3923	+	6.00000i	−0.643268	+	0.371391i
$262$		0			0
$263$	−12.1244	−	7.00000i	−0.747620	−	0.431638i	0.0772134	−	0.997015i	$-0.475398\pi$
$263$	−12.1244	−	7.00000i	−0.747620	−	0.431638i	−0.824833	+	0.565376i	$0.808731\pi$
$264$		0			0
$265$		−	9.79796i		−	0.601884i
$266$		0			0
$267$	−36.0000			−2.20316
$268$		0			0
$269$	6.12372	−	10.6066i	0.373370	−	0.646696i	−0.616712	−	0.787189i	$-0.711536\pi$
$269$	6.12372	−	10.6066i	0.373370	−	0.646696i	0.990082	+	0.140493i	$0.0448688\pi$
$270$		0			0
$271$	−9.79796	−	16.9706i	−0.595184	−	1.03089i	−0.993521	−	0.113649i	$-0.963746\pi$
$271$	−9.79796	−	16.9706i	−0.595184	−	1.03089i	0.398337	−	0.917239i	$-0.369587\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.00000	+	1.73205i	0.0603023	+	0.104447i
$276$		0			0
$277$	10.3923	+	6.00000i	0.624413	+	0.360505i	0.778585	−	0.627539i	$-0.215938\pi$
$277$	10.3923	+	6.00000i	0.624413	+	0.360505i	−0.154172	+	0.988044i	$0.549271\pi$
$278$		0			0
$279$	14.6969			0.879883
$280$		0			0
$281$	2.00000			0.119310			0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000			0.119310			0.0596550	+	0.998219i	$0.481000\pi$
$282$		0			0
$283$	2.12132	+	1.22474i	0.126099	+	0.0728035i	0.561723	−	0.827325i	$-0.310139\pi$
$283$	2.12132	+	1.22474i	0.126099	+	0.0728035i	−0.435623	+	0.900129i	$0.643472\pi$
$284$		0			0
$285$	−7.34847	−	12.7279i	−0.435286	−	0.753937i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	3.50000	+	6.06218i	0.205882	+	0.356599i
$290$		0			0
$291$	6.00000	−	10.3923i	0.351726	−	0.609208i
$292$		0			0
$293$	−2.44949			−0.143101			−0.0715504	−	0.997437i	$-0.522795\pi$
$293$	−2.44949			−0.143101			−0.0715504	+	0.997437i	$0.522795\pi$
$294$		0			0
$295$			6.00000i			0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.48528	+	4.89898i	−0.490716	+	0.283315i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−36.3731	+	21.0000i	−2.08958	+	1.20642i
$304$		0			0
$305$	9.00000	−	15.5885i	0.515339	−	0.892592i
$306$		0			0
$307$		−	31.8434i		−	1.81740i	−0.417453	−	0.908698i	$-0.637077\pi$
$307$		−	31.8434i		−	1.81740i	0.417453	−	0.908698i	$-0.362923\pi$
$308$		0			0
$309$			24.0000i			1.36531i
$310$		0			0
$311$	2.44949	−	4.24264i	0.138898	−	0.240578i	−0.788182	−	0.615442i	$-0.788977\pi$
$311$	2.44949	−	4.24264i	0.138898	−	0.240578i	0.927080	+	0.374864i	$0.122311\pi$
$312$		0			0
$313$	−8.48528	+	4.89898i	−0.479616	+	0.276907i	−0.720257	−	0.693708i	$-0.755976\pi$
$313$	−8.48528	+	4.89898i	−0.479616	+	0.276907i	0.240640	+	0.970614i	$0.422643\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−27.7128	+	16.0000i	−1.55651	+	0.898650i	−0.558920	+	0.829222i	$0.688784\pi$
$317$	−27.7128	+	16.0000i	−1.55651	+	0.898650i	−0.997587	+	0.0694277i	$0.977883\pi$
$318$		0			0
$319$	6.92820	+	4.00000i	0.387905	+	0.223957i
$320$		0			0
$321$		−	4.89898i		−	0.273434i
$322$		0			0
$323$	12.0000			0.667698
$324$		0			0
$325$	1.22474	−	2.12132i	0.0679366	−	0.117670i
$326$		0			0
$327$	−4.89898	−	8.48528i	−0.270914	−	0.469237i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−9.00000	−	15.5885i	−0.494685	−	0.856819i	0.505296	−	0.862946i	$-0.331383\pi$
$331$	−9.00000	−	15.5885i	−0.494685	−	0.856819i	−0.999981	+	0.00612670i	$0.998050\pi$
$332$		0			0
$333$	20.7846	+	12.0000i	1.13899	+	0.657596i
$334$		0			0
$335$	4.89898			0.267660
$336$		0			0
$337$	−12.0000			−0.653682			−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000			−0.653682			−0.326841	+	0.945079i	$0.605984\pi$
$338$		0			0
$339$	8.48528	+	4.89898i	0.460857	+	0.266076i
$340$		0			0
$341$	−4.89898	−	8.48528i	−0.265295	−	0.459504i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−12.0000	−	20.7846i	−0.646058	−	1.11901i
$346$		0			0
$347$	−11.0000	+	19.0526i	−0.590511	+	1.02279i	0.403653	+	0.914912i	$0.367740\pi$
$347$	−11.0000	+	19.0526i	−0.590511	+	1.02279i	−0.994164	+	0.107883i	$0.965593\pi$
$348$		0			0
$349$	12.2474			0.655591			0.327795	−	0.944749i	$-0.393694\pi$
$349$	12.2474			0.655591			0.327795	+	0.944749i	$0.393694\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.48528	+	4.89898i	0.451626	+	0.260746i	0.708517	−	0.705694i	$-0.249365\pi$
$353$	8.48528	+	4.89898i	0.451626	+	0.260746i	−0.256891	+	0.966440i	$0.582698\pi$
$354$		0			0
$355$	−21.2132	+	12.2474i	−1.12588	+	0.650027i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.46410	+	2.00000i	−0.182828	+	0.105556i	−0.588621	−	0.808409i	$-0.700329\pi$
$359$	−3.46410	+	2.00000i	−0.182828	+	0.105556i	0.405793	+	0.913965i	$0.366996\pi$
$360$		0			0
$361$	−6.50000	+	11.2583i	−0.342105	+	0.592544i
$362$		0			0
$363$			17.1464i			0.899954i
$364$		0			0
$365$			36.0000i			1.88433i
$366$		0			0
$367$	−14.6969	+	25.4558i	−0.767174	+	1.32878i	0.171916	+	0.985112i	$0.445004\pi$
$367$	−14.6969	+	25.4558i	−0.767174	+	1.32878i	−0.939090	+	0.343673i	$0.888329\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	31.1769	−	18.0000i	1.61428	−	0.932005i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	31.1769	−	18.0000i	1.61428	−	0.932005i	0.988363	−	0.152115i	$-0.0486083\pi$
$374$		0			0
$375$	−20.7846	−	12.0000i	−1.07331	−	0.619677i
$376$		0			0
$377$		−	9.79796i		−	0.504621i
$378$		0			0
$379$	−30.0000			−1.54100			−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000			−1.54100			−0.770498	+	0.637442i	$0.779993\pi$
$380$		0			0
$381$	−14.6969	+	25.4558i	−0.752947	+	1.30414i
$382$		0			0
$383$	17.1464	+	29.6985i	0.876142	+	1.51752i	0.855542	+	0.517734i	$0.173224\pi$
$383$	17.1464	+	29.6985i	0.876142	+	1.51752i	0.0205998	+	0.999788i	$0.493442\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	9.00000	+	15.5885i	0.457496	+	0.792406i
$388$		0			0
$389$	13.8564	+	8.00000i	0.702548	+	0.405616i	0.808296	−	0.588777i	$-0.200390\pi$
$389$	13.8564	+	8.00000i	0.702548	+	0.405616i	−0.105748	+	0.994393i	$0.533724\pi$
$390$		0			0
$391$	19.5959			0.991008
$392$		0			0
$393$	−30.0000			−1.51330
$394$		0			0
$395$	−12.7279	−	7.34847i	−0.640411	−	0.369742i
$396$		0			0
$397$	−15.9217	−	27.5772i	−0.799086	−	1.38406i	−0.920212	−	0.391421i	$-0.871984\pi$
$397$	−15.9217	−	27.5772i	−0.799086	−	1.38406i	0.121125	−	0.992637i	$-0.461350\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−16.0000	−	27.7128i	−0.799002	−	1.38391i	−0.920267	−	0.391292i	$-0.872028\pi$
$401$	−16.0000	−	27.7128i	−0.799002	−	1.38391i	0.121265	−	0.992620i	$-0.461305\pi$
$402$		0			0
$403$	−6.00000	+	10.3923i	−0.298881	+	0.517678i
$404$		0			0
$405$	−22.0454			−1.09545
$406$		0			0
$407$		−	16.0000i		−	0.793091i
$408$		0			0
$409$	−8.48528	−	4.89898i	−0.419570	−	0.242239i	0.275323	−	0.961352i	$-0.411215\pi$
$409$	−8.48528	−	4.89898i	−0.419570	−	0.242239i	−0.694893	+	0.719113i	$0.744548\pi$
$410$		0			0
$411$	4.24264	−	2.44949i	0.209274	−	0.120824i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−5.19615	+	3.00000i	−0.255069	+	0.147264i
$416$		0			0
$417$	−3.00000	+	5.19615i	−0.146911	+	0.254457i
$418$		0			0
$419$			26.9444i			1.31632i	0.752878	+	0.658160i	$0.228665\pi$
$419$			26.9444i			1.31632i	−0.752878	+	0.658160i	$0.771335\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	7.34847	−	12.7279i	0.357295	−	0.618853i
$424$		0			0
$425$	−4.24264	+	2.44949i	−0.205798	+	0.118818i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−10.3923	+	6.00000i	−0.501745	+	0.289683i
$430$		0			0
$431$	−17.3205	−	10.0000i	−0.834300	−	0.481683i	0.0210230	−	0.999779i	$-0.493308\pi$
$431$	−17.3205	−	10.0000i	−0.834300	−	0.481683i	−0.855323	+	0.518096i	$0.826641\pi$
$432$		0			0
$433$			14.6969i			0.706290i	0.935569	+	0.353145i	$0.114888\pi$
$433$			14.6969i			0.706290i	−0.935569	+	0.353145i	$0.885112\pi$
$434$		0			0
$435$	24.0000			1.15071
$436$		0			0
$437$	4.89898	−	8.48528i	0.234350	−	0.405906i
$438$		0			0
$439$	12.2474	+	21.2132i	0.584539	+	1.01245i	0.994933	+	0.100543i	$0.0320579\pi$
$439$	12.2474	+	21.2132i	0.584539	+	1.01245i	−0.410394	+	0.911908i	$0.634609\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.0000	−	22.5167i	−0.617649	−	1.06980i	−0.989914	−	0.141672i	$-0.954752\pi$
$443$	−13.0000	−	22.5167i	−0.617649	−	1.06980i	0.372265	−	0.928126i	$-0.378581\pi$
$444$		0			0
$445$	31.1769	+	18.0000i	1.47793	+	0.853282i
$446$		0			0
$447$	−39.1918			−1.85371
$448$		0			0
$449$	−10.0000			−0.471929			−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000			−0.471929			−0.235965	+	0.971762i	$0.575825\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	24.4949	+	42.4264i	1.15087	+	1.99337i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.00000	+	10.3923i	0.280668	+	0.486132i	0.971549	−	0.236837i	$-0.0761106\pi$
$457$	6.00000	+	10.3923i	0.280668	+	0.486132i	−0.690881	+	0.722968i	$0.742777\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.34847			−0.342252			−0.171126	−	0.985249i	$-0.554741\pi$
$461$	−7.34847			−0.342252			−0.171126	+	0.985249i	$0.554741\pi$
$462$		0			0
$463$		−	6.00000i		−	0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$		−	6.00000i		−	0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$		0			0
$465$	−25.4558	−	14.6969i	−1.18049	−	0.681554i
$466$		0			0
$467$	36.0624	−	20.8207i	1.66877	−	0.963465i	0.700469	−	0.713683i	$-0.252974\pi$
$467$	36.0624	−	20.8207i	1.66877	−	0.963465i	0.968302	−	0.249783i	$-0.0803591\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−15.5885	+	9.00000i	−0.718278	+	0.414698i
$472$		0			0
$473$	6.00000	−	10.3923i	0.275880	−	0.477839i
$474$		0			0
$475$			2.44949i			0.112390i
$476$		0			0
$477$			12.0000i			0.549442i
$478$		0			0
$479$	−12.2474	+	21.2132i	−0.559600	+	0.969256i	0.437929	+	0.899009i	$0.355712\pi$
$479$	−12.2474	+	21.2132i	−0.559600	+	0.969256i	−0.997530	+	0.0702467i	$0.977621\pi$
$480$		0			0
$481$	−16.9706	+	9.79796i	−0.773791	+	0.446748i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.3923	+	6.00000i	−0.471890	+	0.272446i
$486$		0			0
$487$	24.2487	+	14.0000i	1.09881	+	0.634401i	0.935909	−	0.352241i	$-0.114580\pi$
$487$	24.2487	+	14.0000i	1.09881	+	0.634401i	0.162905	+	0.986642i	$0.447914\pi$
$488$		0			0
$489$			34.2929i			1.55078i
$490$		0			0
$491$	−2.00000			−0.0902587			−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000			−0.0902587			−0.0451294	+	0.998981i	$0.514370\pi$
$492$		0			0
$493$	−9.79796	+	16.9706i	−0.441278	+	0.764316i
$494$		0			0
$495$	−7.34847	−	12.7279i	−0.330289	−	0.572078i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	15.0000	+	25.9808i	0.671492	+	1.16306i	0.977481	+	0.211024i	$0.0676797\pi$
$499$	15.0000	+	25.9808i	0.671492	+	1.16306i	−0.305989	+	0.952035i	$0.598987\pi$
$500$		0			0
$501$	−41.5692	−	24.0000i	−1.85718	−	1.07224i
$502$		0			0
$503$	14.6969			0.655304			0.327652	−	0.944798i	$-0.393743\pi$
$503$	14.6969			0.655304			0.327652	+	0.944798i	$0.393743\pi$
$504$		0			0
$505$	42.0000			1.86898
$506$		0			0
$507$	−14.8492	−	8.57321i	−0.659478	−	0.380750i
$508$		0			0
$509$	18.3712	+	31.8198i	0.814288	+	1.41039i	0.909838	+	0.414963i	$0.136206\pi$
$509$	18.3712	+	31.8198i	0.814288	+	1.41039i	−0.0955502	+	0.995425i	$0.530461\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.0000	−	20.7846i	0.528783	−	0.915879i
$516$		0			0
$517$	−9.79796			−0.430914
$518$		0			0
$519$			6.00000i			0.263371i
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−23.3345	+	13.4722i	−1.02035	+	0.589098i	−0.914205	−	0.405253i	$-0.867183\pi$
$523$	−23.3345	+	13.4722i	−1.02035	+	0.589098i	−0.106143	+	0.994351i	$0.533850\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	20.7846	−	12.0000i	0.905392	−	0.522728i
$528$		0			0
$529$	−3.50000	+	6.06218i	−0.152174	+	0.263573i
$530$		0			0
$531$		−	7.34847i		−	0.318896i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−2.44949	+	4.24264i	−0.105901	+	0.183425i
$536$		0			0
$537$	21.2132	−	12.2474i	0.915417	−	0.528516i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$	−15.5885	−	9.00000i	−0.668965	−	0.386227i
$544$		0			0
$545$			9.79796i			0.419698i
$546$		0			0
$547$	2.00000			0.0855138			0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000			0.0855138			0.0427569	+	0.999086i	$0.486386\pi$
$548$		0			0
$549$	−11.0227	+	19.0919i	−0.470438	+	0.814822i
$550$		0			0
$551$	4.89898	+	8.48528i	0.208704	+	0.361485i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−24.0000	−	41.5692i	−1.01874	−	1.76452i
$556$		0			0
$557$	−24.2487	−	14.0000i	−1.02745	−	0.593199i	−0.111198	−	0.993798i	$-0.535469\pi$
$557$	−24.2487	−	14.0000i	−1.02745	−	0.593199i	−0.916253	+	0.400599i	$0.868802\pi$
$558$		0			0
$559$	−14.6969			−0.621614
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$	−19.0919	−	11.0227i	−0.804627	−	0.464552i	0.0404596	−	0.999181i	$-0.487118\pi$
$563$	−19.0919	−	11.0227i	−0.804627	−	0.464552i	−0.845087	+	0.534630i	$0.820451\pi$
$564$		0			0
$565$	−4.89898	−	8.48528i	−0.206102	−	0.356978i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.0000	+	17.3205i	0.419222	+	0.726113i	0.995861	−	0.0908852i	$-0.0289696\pi$
$569$	10.0000	+	17.3205i	0.419222	+	0.726113i	−0.576640	+	0.816999i	$0.695636\pi$
$570$		0			0
$571$	1.00000	−	1.73205i	0.0418487	−	0.0724841i	−0.844342	−	0.535804i	$-0.820009\pi$
$571$	1.00000	−	1.73205i	0.0418487	−	0.0724841i	0.886191	+	0.463320i	$0.153342\pi$
$572$		0			0
$573$	24.4949			1.02329
$574$		0			0
$575$			4.00000i			0.166812i
$576$		0			0
$577$	−16.9706	−	9.79796i	−0.706494	−	0.407894i	0.103268	−	0.994654i	$-0.467070\pi$
$577$	−16.9706	−	9.79796i	−0.706494	−	0.407894i	−0.809761	+	0.586759i	$0.800404\pi$
$578$		0			0
$579$	−50.9117	+	29.3939i	−2.11582	+	1.22157i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	6.92820	−	4.00000i	0.286937	−	0.165663i
$584$		0			0
$585$	−9.00000	+	15.5885i	−0.372104	+	0.644503i
$586$		0			0
$587$		−	7.34847i		−	0.303304i	−0.988434	−	0.151652i	$-0.951541\pi$
$587$		−	7.34847i		−	0.303304i	0.988434	−	0.151652i	$-0.0484593\pi$
$588$		0			0
$589$		−	12.0000i		−	0.494451i
$590$		0			0
$591$	−9.79796	+	16.9706i	−0.403034	+	0.698076i
$592$		0			0
$593$	−8.48528	+	4.89898i	−0.348449	+	0.201177i	−0.664002	−	0.747731i	$-0.731143\pi$
$593$	−8.48528	+	4.89898i	−0.348449	+	0.201177i	0.315553	+	0.948908i	$0.397810\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.1244	+	7.00000i	0.495388	+	0.286012i	0.726807	−	0.686842i	$-0.241004\pi$
$599$	12.1244	+	7.00000i	0.495388	+	0.286012i	−0.231419	+	0.972854i	$0.574337\pi$
$600$		0			0
$601$		−	24.4949i		−	0.999168i	−0.866266	−	0.499584i	$-0.833486\pi$
$601$		−	24.4949i		−	0.999168i	0.866266	−	0.499584i	$-0.166514\pi$
$602$		0			0
$603$	−6.00000			−0.244339
$604$		0			0
$605$	8.57321	−	14.8492i	0.348551	−	0.603708i
$606$		0			0
$607$	−14.6969	−	25.4558i	−0.596530	−	1.03322i	−0.993329	−	0.115315i	$-0.963212\pi$
$607$	−14.6969	−	25.4558i	−0.596530	−	1.03322i	0.396799	−	0.917906i	$-0.370121\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.00000	+	10.3923i	0.242734	+	0.420428i
$612$		0			0
$613$	20.7846	+	12.0000i	0.839482	+	0.484675i	0.857088	−	0.515170i	$-0.172271\pi$
$613$	20.7846	+	12.0000i	0.839482	+	0.484675i	−0.0176058	+	0.999845i	$0.505604\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−32.0000			−1.28827			−0.644136	−	0.764911i	$-0.722783\pi$
$617$	−32.0000			−1.28827			−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$	19.0919	+	11.0227i	0.767368	+	0.443040i	0.831935	−	0.554873i	$-0.187233\pi$
$619$	19.0919	+	11.0227i	0.767368	+	0.443040i	−0.0645672	+	0.997913i	$0.520567\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		0			0
$627$	6.00000	−	10.3923i	0.239617	−	0.415029i
$628$		0			0
$629$	39.1918			1.56268
$630$		0			0
$631$		−	30.0000i		−	1.19428i	−0.802137	−	0.597141i	$-0.796303\pi$
$631$		−	30.0000i		−	1.19428i	0.802137	−	0.597141i	$-0.203697\pi$
$632$		0			0
$633$	38.1838	+	22.0454i	1.51767	+	0.876226i
$634$		0			0
$635$	25.4558	−	14.6969i	1.01018	−	0.583230i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	25.9808	−	15.0000i	1.02778	−	0.593391i
$640$		0			0
$641$	4.00000	−	6.92820i	0.157991	−	0.273648i	−0.776153	−	0.630544i	$-0.782832\pi$
$641$	4.00000	−	6.92820i	0.157991	−	0.273648i	0.934144	+	0.356897i	$0.116165\pi$
$642$		0			0
$643$			22.0454i			0.869386i	0.900579	+	0.434693i	$0.143143\pi$
$643$			22.0454i			0.869386i	−0.900579	+	0.434693i	$0.856857\pi$
$644$		0			0
$645$		−	36.0000i		−	1.41750i
$646$		0			0
$647$	9.79796	−	16.9706i	0.385198	−	0.667182i	−0.606599	−	0.795008i	$-0.707467\pi$
$647$	9.79796	−	16.9706i	0.385198	−	0.667182i	0.991797	+	0.127826i	$0.0408000\pi$
$648$		0			0
$649$	−4.24264	+	2.44949i	−0.166538	+	0.0961509i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.8564	−	8.00000i	0.542243	−	0.313064i	−0.203744	−	0.979024i	$-0.565311\pi$
$653$	13.8564	−	8.00000i	0.542243	−	0.313064i	0.745988	+	0.665960i	$0.231978\pi$
$654$		0			0
$655$	25.9808	+	15.0000i	1.01515	+	0.586098i
$656$		0			0
$657$		−	44.0908i		−	1.72015i
$658$		0			0
$659$	−10.0000			−0.389545			−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000			−0.389545			−0.194772	+	0.980848i	$0.562397\pi$
$660$		0			0
$661$	15.9217	−	27.5772i	0.619282	−	1.07263i	−0.370335	−	0.928898i	$-0.620757\pi$
$661$	15.9217	−	27.5772i	0.619282	−	1.07263i	0.989617	−	0.143729i	$-0.0459094\pi$
$662$		0			0
$663$	−14.6969	−	25.4558i	−0.570782	−	0.988623i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8.00000	+	13.8564i	0.309761	+	0.536522i
$668$		0			0
$669$	−20.7846	−	12.0000i	−0.803579	−	0.463947i
$670$		0			0
$671$	14.6969			0.567369
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.9217	−	27.5772i	−0.611920	−	1.05988i	−0.990917	−	0.134478i	$-0.957064\pi$
$677$	−15.9217	−	27.5772i	−0.611920	−	1.05988i	0.378997	−	0.925398i	$-0.376269\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	9.00000	+	15.5885i	0.344881	+	0.597351i
$682$		0			0
$683$	7.00000	−	12.1244i	0.267848	−	0.463926i	−0.700458	−	0.713693i	$-0.747021\pi$
$683$	7.00000	−	12.1244i	0.267848	−	0.463926i	0.968306	+	0.249768i	$0.0803543\pi$
$684$		0			0
$685$	−4.89898			−0.187180
$686$		0			0
$687$			30.0000i			1.14457i
$688$		0			0
$689$	−8.48528	−	4.89898i	−0.323263	−	0.186636i
$690$		0			0
$691$	31.8198	−	18.3712i	1.21048	−	0.698872i	0.247618	−	0.968858i	$-0.420352\pi$
$691$	31.8198	−	18.3712i	1.21048	−	0.698872i	0.962864	+	0.269985i	$0.0870189\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.19615	−	3.00000i	0.197101	−	0.113796i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	34.2929i		−	1.29707i
$700$		0			0
$701$		−	40.0000i		−	1.51078i	−0.655276	−	0.755390i	$-0.727448\pi$
$701$		−	40.0000i		−	1.51078i	0.655276	−	0.755390i	$-0.272552\pi$
$702$		0			0
$703$	9.79796	−	16.9706i	0.369537	−	0.640057i
$704$		0			0
$705$	−25.4558	+	14.6969i	−0.958723	+	0.553519i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	3.46410	−	2.00000i	0.130097	−	0.0751116i	−0.433539	−	0.901135i	$-0.642735\pi$
$709$	3.46410	−	2.00000i	0.130097	−	0.0751116i	0.563636	+	0.826023i	$0.309402\pi$
$710$		0			0
$711$	15.5885	+	9.00000i	0.584613	+	0.337526i
$712$		0			0
$713$		−	19.5959i		−	0.733873i
$714$		0			0
$715$	12.0000			0.448775
$716$		0			0
$717$	4.89898	−	8.48528i	0.182956	−	0.316889i
$718$		0			0
$719$	12.2474	+	21.2132i	0.456753	+	0.791119i	0.998787	−	0.0492373i	$-0.0156791\pi$
$719$	12.2474	+	21.2132i	0.456753	+	0.791119i	−0.542034	+	0.840356i	$0.682346\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	30.0000	+	51.9615i	1.11571	+	1.93247i
$724$		0			0
$725$	−3.46410	−	2.00000i	−0.128654	−	0.0742781i
$726$		0			0
$727$	29.3939			1.09016			0.545079	−	0.838385i	$-0.316500\pi$
$727$	29.3939			1.09016			0.545079	+	0.838385i	$0.316500\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	25.4558	+	14.6969i	0.941518	+	0.543586i
$732$		0			0
$733$	−11.0227	−	19.0919i	−0.407133	−	0.705175i	0.587434	−	0.809272i	$-0.300138\pi$
$733$	−11.0227	−	19.0919i	−0.407133	−	0.705175i	−0.994567	+	0.104097i	$0.966805\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.00000	+	3.46410i	0.0736709	+	0.127602i
$738$		0			0
$739$	−25.0000	+	43.3013i	−0.919640	+	1.59286i	−0.119677	+	0.992813i	$0.538186\pi$
$739$	−25.0000	+	43.3013i	−0.919640	+	1.59286i	−0.799962	+	0.600050i	$0.795147\pi$
$740$		0			0
$741$	−14.6969			−0.539906
$742$		0			0
$743$			44.0000i			1.61420i	0.590412	+	0.807102i	$0.298965\pi$
$743$			44.0000i			1.61420i	−0.590412	+	0.807102i	$0.701035\pi$
$744$		0			0
$745$	33.9411	+	19.5959i	1.24351	+	0.717939i
$746$		0			0
$747$	6.36396	−	3.67423i	0.232845	−	0.134433i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	17.3205	−	10.0000i	0.632034	−	0.364905i	−0.149505	−	0.988761i	$-0.547768\pi$
$751$	17.3205	−	10.0000i	0.632034	−	0.364905i	0.781540	+	0.623856i	$0.214435\pi$
$752$		0			0
$753$	15.0000	−	25.9808i	0.546630	−	0.946792i
$754$		0			0
$755$		−	48.9898i		−	1.78292i
$756$		0			0
$757$			8.00000i			0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$			8.00000i			0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$		0			0
$759$	9.79796	−	16.9706i	0.355643	−	0.615992i
$760$		0			0
$761$	42.4264	−	24.4949i	1.53796	−	0.887939i	0.538998	−	0.842307i	$-0.318803\pi$
$761$	42.4264	−	24.4949i	1.53796	−	0.887939i	0.998958	−	0.0456321i	$-0.0145302\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	31.1769	−	18.0000i	1.12720	−	0.650791i
$766$		0			0
$767$	5.19615	+	3.00000i	0.187622	+	0.108324i
$768$		0			0
$769$			34.2929i			1.23663i	0.785930	+	0.618316i	$0.212185\pi$
$769$			34.2929i			1.23663i	−0.785930	+	0.618316i	$0.787815\pi$
$770$		0			0
$771$	48.0000			1.72868
$772$		0			0
$773$	−11.0227	+	19.0919i	−0.396459	+	0.686687i	−0.993286	−	0.115683i	$-0.963094\pi$
$773$	−11.0227	+	19.0919i	−0.396459	+	0.686687i	0.596827	+	0.802370i	$0.296428\pi$
$774$		0			0
$775$	2.44949	+	4.24264i	0.0879883	+	0.152400i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−17.3205	−	10.0000i	−0.619777	−	0.357828i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.0000			0.642448
$786$		0			0
$787$	6.36396	+	3.67423i	0.226851	+	0.130972i	0.609118	−	0.793079i	$-0.291523\pi$
$787$	6.36396	+	3.67423i	0.226851	+	0.130972i	−0.382268	+	0.924052i	$0.624857\pi$
$788$		0			0
$789$	−17.1464	−	29.6985i	−0.610429	−	1.05729i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−9.00000	−	15.5885i	−0.319599	−	0.553562i
$794$		0			0
$795$	12.0000	−	20.7846i	0.425596	−	0.737154i
$796$		0			0
$797$	−41.6413			−1.47501			−0.737506	−	0.675341i	$-0.763997\pi$
$797$	−41.6413			−1.47501			−0.737506	+	0.675341i	$0.763997\pi$
$798$		0			0
$799$		−	24.0000i		−	0.849059i
$800$		0			0
$801$	−38.1838	−	22.0454i	−1.34916	−	0.778936i
$802$		0			0
$803$	−25.4558	+	14.6969i	−0.898317	+	0.518644i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	25.9808	−	15.0000i	0.914566	−	0.528025i
$808$		0			0
$809$	−10.0000	+	17.3205i	−0.351581	+	0.608957i	−0.986527	−	0.163600i	$-0.947689\pi$
$809$	−10.0000	+	17.3205i	−0.351581	+	0.608957i	0.634945	+	0.772557i	$0.281023\pi$
$810$		0			0
$811$			36.7423i			1.29020i	0.764099	+	0.645099i	$0.223184\pi$
$811$			36.7423i			1.29020i	−0.764099	+	0.645099i	$0.776816\pi$
$812$		0			0
$813$		−	48.0000i		−	1.68343i
$814$		0			0
$815$	17.1464	−	29.6985i	0.600613	−	1.04029i
$816$		0			0
$817$	12.7279	−	7.34847i	0.445294	−	0.257090i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−17.3205	+	10.0000i	−0.604490	+	0.349002i	−0.770806	−	0.637070i	$-0.780146\pi$
$821$	−17.3205	+	10.0000i	−0.604490	+	0.349002i	0.166316	+	0.986073i	$0.446813\pi$
$822$		0			0
$823$	−46.7654	−	27.0000i	−1.63014	−	0.941161i	−0.984049	−	0.177899i	$-0.943070\pi$
$823$	−46.7654	−	27.0000i	−1.63014	−	0.941161i	−0.646090	−	0.763261i	$-0.723597\pi$
$824$		0			0
$825$			4.89898i			0.170561i
$826$		0			0
$827$	22.0000			0.765015			0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000			0.765015			0.382507	+	0.923952i	$0.375061\pi$
$828$		0			0
$829$	18.3712	−	31.8198i	0.638057	−	1.10515i	−0.347801	−	0.937568i	$-0.613072\pi$
$829$	18.3712	−	31.8198i	0.638057	−	1.10515i	0.985859	−	0.167579i	$-0.0535950\pi$
$830$		0			0
$831$	14.6969	+	25.4558i	0.509831	+	0.883053i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	24.0000	+	41.5692i	0.830554	+	1.43856i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$		0			0
$843$	4.24264	+	2.44949i	0.146124	+	0.0843649i
$844$		0			0
$845$	8.57321	+	14.8492i	0.294928	+	0.510829i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	3.00000	+	5.19615i	0.102960	+	0.178331i
$850$		0			0
$851$	16.0000	−	27.7128i	0.548473	−	0.949983i
$852$		0			0
$853$	−2.44949			−0.0838689			−0.0419345	−	0.999120i	$-0.513352\pi$
$853$	−2.44949			−0.0838689			−0.0419345	+	0.999120i	$0.513352\pi$
$854$		0			0
$855$		−	18.0000i		−	0.615587i
$856$		0			0
$857$	25.4558	+	14.6969i	0.869555	+	0.502038i	0.867200	−	0.497959i	$-0.165917\pi$
$857$	25.4558	+	14.6969i	0.869555	+	0.502038i	0.00235471	+	0.999997i	$0.499250\pi$
$858$		0			0
$859$	23.3345	−	13.4722i	0.796164	−	0.459665i	−0.0459643	−	0.998943i	$-0.514636\pi$
$859$	23.3345	−	13.4722i	0.796164	−	0.459665i	0.842128	+	0.539278i	$0.181303\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−39.8372	+	23.0000i	−1.35607	+	0.782929i	−0.989092	−	0.147299i	$-0.952942\pi$
$863$	−39.8372	+	23.0000i	−1.35607	+	0.782929i	−0.366981	+	0.930228i	$0.619609\pi$
$864$		0			0
$865$	3.00000	−	5.19615i	0.102003	−	0.176674i
$866$		0			0
$867$			17.1464i			0.582323i
$868$		0			0
$869$		−	12.0000i		−	0.407072i
$870$		0			0
$871$	2.44949	−	4.24264i	0.0829978	−	0.143756i
$872$		0			0
$873$	12.7279	−	7.34847i	0.430775	−	0.248708i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	41.5692	−	24.0000i	1.40369	−	0.810422i	0.408923	−	0.912569i	$-0.365904\pi$
$877$	41.5692	−	24.0000i	1.40369	−	0.810422i	0.994769	+	0.102146i	$0.0325710\pi$
$878$		0			0
$879$	−5.19615	−	3.00000i	−0.175262	−	0.101187i
$880$		0			0
$881$			48.9898i			1.65051i	0.564762	+	0.825254i	$0.308968\pi$
$881$			48.9898i			1.65051i	−0.564762	+	0.825254i	$0.691032\pi$
$882$		0			0
$883$	6.00000			0.201916			0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000			0.201916			0.100958	+	0.994891i	$0.467809\pi$
$884$		0			0
$885$	−7.34847	+	12.7279i	−0.247016	+	0.427844i
$886$		0			0
$887$	−14.6969	−	25.4558i	−0.493475	−	0.854724i	0.506497	−	0.862242i	$-0.330940\pi$
$887$	−14.6969	−	25.4558i	−0.493475	−	0.854724i	−0.999972	+	0.00751822i	$0.997607\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−9.00000	−	15.5885i	−0.301511	−	0.522233i
$892$		0			0
$893$	−10.3923	−	6.00000i	−0.347765	−	0.200782i
$894$		0			0
$895$	−24.4949			−0.818774
$896$		0			0
$897$	−24.0000			−0.801337
$898$		0			0
$899$	16.9706	+	9.79796i	0.566000	+	0.326780i
$900$		0			0
$901$	9.79796	+	16.9706i	0.326417	+	0.565371i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.00000	+	15.5885i	0.299170	+	0.518178i
$906$		0			0
$907$	−21.0000	+	36.3731i	−0.697294	+	1.20775i	0.272108	+	0.962267i	$0.412279\pi$
$907$	−21.0000	+	36.3731i	−0.697294	+	1.20775i	−0.969401	+	0.245481i	$0.921054\pi$
$908$		0			0
$909$	−51.4393			−1.70613
$910$		0			0
$911$		−	20.0000i		−	0.662630i	−0.943520	−	0.331315i	$-0.892508\pi$
$911$		−	20.0000i		−	0.662630i	0.943520	−	0.331315i	$-0.107492\pi$
$912$		0			0
$913$	−4.24264	−	2.44949i	−0.140411	−	0.0810663i
$914$		0			0
$915$	38.1838	−	22.0454i	1.26232	−	0.728799i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	39.8372	−	23.0000i	1.31411	−	0.758700i	0.331333	−	0.943514i	$-0.392502\pi$
$919$	39.8372	−	23.0000i	1.31411	−	0.758700i	0.982774	+	0.184814i	$0.0591682\pi$
$920$		0			0
$921$	39.0000	−	67.5500i	1.28509	−	2.22585i
$922$		0			0
$923$			24.4949i			0.806259i
$924$		0			0
$925$			8.00000i			0.263038i
$926$		0			0
$927$	−14.6969	+	25.4558i	−0.482711	+	0.836080i
$928$		0			0
$929$	−12.7279	+	7.34847i	−0.417590	+	0.241095i	−0.694045	−	0.719931i	$-0.744173\pi$
$929$	−12.7279	+	7.34847i	−0.417590	+	0.241095i	0.276456	+	0.961027i	$0.410840\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	10.3923	−	6.00000i	0.340229	−	0.196431i
$934$		0			0
$935$	−20.7846	−	12.0000i	−0.679729	−	0.392442i
$936$		0			0
$937$		−	4.89898i		−	0.160043i	−0.996793	−	0.0800213i	$-0.974501\pi$
$937$		−	4.89898i		−	0.160043i	0.996793	−	0.0800213i	$-0.0254988\pi$
$938$		0			0
$939$	−24.0000			−0.783210
$940$		0			0
$941$	15.9217	−	27.5772i	0.519032	−	0.898990i	−0.480723	−	0.876872i	$-0.659626\pi$
$941$	15.9217	−	27.5772i	0.519032	−	0.898990i	0.999755	−	0.0221175i	$-0.00704081\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.0000	−	19.0526i	−0.357452	−	0.619125i	0.630082	−	0.776528i	$-0.283021\pi$
$947$	−11.0000	−	19.0526i	−0.357452	−	0.619125i	−0.987534	+	0.157403i	$0.949688\pi$
$948$		0			0
$949$	31.1769	+	18.0000i	1.01205	+	0.584305i
$950$		0			0
$951$	−78.3837			−2.54176
$952$		0			0
$953$	34.0000			1.10137			0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000			1.10137			0.550684	+	0.834714i	$0.314367\pi$
$954$		0			0
$955$	−21.2132	−	12.2474i	−0.686443	−	0.396318i
$956$		0			0
$957$	9.79796	+	16.9706i	0.316723	+	0.548580i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.50000	+	6.06218i	0.112903	+	0.195554i
$962$		0			0
$963$	3.00000	−	5.19615i	0.0966736	−	0.167444i
$964$		0			0
$965$	58.7878			1.89244
$966$		0			0
$967$			12.0000i			0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$			12.0000i			0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$		0			0
$969$	25.4558	+	14.6969i	0.817760	+	0.472134i
$970$		0			0
$971$	−31.8198	+	18.3712i	−1.02115	+	0.589559i	−0.914436	−	0.404731i	$-0.867365\pi$
$971$	−31.8198	+	18.3712i	−1.02115	+	0.589559i	−0.106710	+	0.994290i	$0.534032\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	5.19615	−	3.00000i	0.166410	−	0.0960769i
$976$		0			0
$977$	−19.0000	+	32.9090i	−0.607864	+	1.05285i	0.383728	+	0.923446i	$0.374640\pi$
$977$	−19.0000	+	32.9090i	−0.607864	+	1.05285i	−0.991592	+	0.129405i	$0.958693\pi$
$978$		0			0
$979$			29.3939i			0.939432i
$980$		0			0
$981$		−	12.0000i		−	0.383131i
$982$		0			0
$983$	−19.5959	+	33.9411i	−0.625013	+	1.08255i	0.363526	+	0.931584i	$0.381573\pi$
$983$	−19.5959	+	33.9411i	−0.625013	+	1.08255i	−0.988538	+	0.150970i	$0.951760\pi$
$984$		0			0
$985$	16.9706	−	9.79796i	0.540727	−	0.312189i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	20.7846	−	12.0000i	0.660912	−	0.381578i
$990$		0			0
$991$	8.66025	+	5.00000i	0.275102	+	0.158830i	0.631204	−	0.775617i	$-0.282561\pi$
$991$	8.66025	+	5.00000i	0.275102	+	0.158830i	−0.356102	+	0.934447i	$0.615894\pi$
$992$		0			0
$993$		−	44.0908i		−	1.39918i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−15.9217	+	27.5772i	−0.504245	+	0.873378i	0.495743	+	0.868469i	$0.334896\pi$
$997$	−15.9217	+	27.5772i	−0.504245	+	0.873378i	−0.999988	+	0.00490839i	$0.998438\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.q.e.815.3		8
4.3	odd	2		392.2.m.f.227.1		8
7.2	even	3	inner	1568.2.q.e.1391.1		8
7.3	odd	6		224.2.e.b.111.3		4
7.4	even	3		224.2.e.b.111.2		4
7.5	odd	6	inner	1568.2.q.e.1391.4		8
7.6	odd	2	inner	1568.2.q.e.815.2		8
8.3	odd	2	inner	1568.2.q.e.815.4		8
8.5	even	2		392.2.m.f.227.3		8
21.11	odd	6		2016.2.p.e.559.2		4
21.17	even	6		2016.2.p.e.559.4		4
28.3	even	6		56.2.e.b.27.1	✓	4
28.11	odd	6		56.2.e.b.27.2	yes	4
28.19	even	6		392.2.m.f.19.3		8
28.23	odd	6		392.2.m.f.19.4		8
28.27	even	2		392.2.m.f.227.2		8
56.3	even	6		224.2.e.b.111.4		4
56.5	odd	6		392.2.m.f.19.1		8
56.11	odd	6		224.2.e.b.111.1		4
56.13	odd	2		392.2.m.f.227.4		8
56.19	even	6	inner	1568.2.q.e.1391.3		8
56.27	even	2	inner	1568.2.q.e.815.1		8
56.37	even	6		392.2.m.f.19.2		8
56.45	odd	6		56.2.e.b.27.3	yes	4
56.51	odd	6	inner	1568.2.q.e.1391.2		8
56.53	even	6		56.2.e.b.27.4	yes	4
84.11	even	6		504.2.p.f.307.3		4
84.59	odd	6		504.2.p.f.307.4		4
112.3	even	12		1792.2.f.e.1791.4		4
112.11	odd	12		1792.2.f.f.1791.4		4
112.45	odd	12		1792.2.f.e.1791.2		4
112.53	even	12		1792.2.f.f.1791.2		4
112.59	even	12		1792.2.f.f.1791.1		4
112.67	odd	12		1792.2.f.e.1791.1		4
112.101	odd	12		1792.2.f.f.1791.3		4
112.109	even	12		1792.2.f.e.1791.3		4
168.11	even	6		2016.2.p.e.559.3		4
168.53	odd	6		504.2.p.f.307.2		4
168.59	odd	6		2016.2.p.e.559.1		4
168.101	even	6		504.2.p.f.307.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.e.b.27.1	✓	4	28.3	even	6
56.2.e.b.27.2	yes	4	28.11	odd	6
56.2.e.b.27.3	yes	4	56.45	odd	6
56.2.e.b.27.4	yes	4	56.53	even	6
224.2.e.b.111.1		4	56.11	odd	6
224.2.e.b.111.2		4	7.4	even	3
224.2.e.b.111.3		4	7.3	odd	6
224.2.e.b.111.4		4	56.3	even	6
392.2.m.f.19.1		8	56.5	odd	6
392.2.m.f.19.2		8	56.37	even	6
392.2.m.f.19.3		8	28.19	even	6
392.2.m.f.19.4		8	28.23	odd	6
392.2.m.f.227.1		8	4.3	odd	2
392.2.m.f.227.2		8	28.27	even	2
392.2.m.f.227.3		8	8.5	even	2
392.2.m.f.227.4		8	56.13	odd	2
504.2.p.f.307.1		4	168.101	even	6
504.2.p.f.307.2		4	168.53	odd	6
504.2.p.f.307.3		4	84.11	even	6
504.2.p.f.307.4		4	84.59	odd	6
1568.2.q.e.815.1		8	56.27	even	2	inner
1568.2.q.e.815.2		8	7.6	odd	2	inner
1568.2.q.e.815.3		8	1.1	even	1	trivial
1568.2.q.e.815.4		8	8.3	odd	2	inner
1568.2.q.e.1391.1		8	7.2	even	3	inner
1568.2.q.e.1391.2		8	56.51	odd	6	inner
1568.2.q.e.1391.3		8	56.19	even	6	inner
1568.2.q.e.1391.4		8	7.5	odd	6	inner
1792.2.f.e.1791.1		4	112.67	odd	12
1792.2.f.e.1791.2		4	112.45	odd	12
1792.2.f.e.1791.3		4	112.109	even	12
1792.2.f.e.1791.4		4	112.3	even	12
1792.2.f.f.1791.1		4	112.59	even	12
1792.2.f.f.1791.2		4	112.53	even	12
1792.2.f.f.1791.3		4	112.101	odd	12
1792.2.f.f.1791.4		4	112.11	odd	12
2016.2.p.e.559.1		4	168.59	odd	6
2016.2.p.e.559.2		4	21.11	odd	6
2016.2.p.e.559.3		4	168.11	even	6
2016.2.p.e.559.4		4	21.17	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.12132 + 1.22474i) q^{3} +(-1.22474 - 2.12132i) q^{5} +(1.50000 + 2.59808i) q^{9} +(1.00000 - 1.73205i) q^{11} -2.44949 q^{13} -6.00000i q^{15} +(4.24264 + 2.44949i) q^{17} +(2.12132 - 1.22474i) q^{19} +(3.46410 - 2.00000i) q^{23} +(-0.500000 + 0.866025i) q^{25} +4.00000i q^{29} +(2.44949 - 4.24264i) q^{31} +(4.24264 - 2.44949i) q^{33} +(6.92820 - 4.00000i) q^{37} +(-5.19615 - 3.00000i) q^{39} +6.00000 q^{43} +(3.67423 - 6.36396i) q^{45} +(-2.44949 - 4.24264i) q^{47} +(6.00000 + 10.3923i) q^{51} +(3.46410 + 2.00000i) q^{53} -4.89898 q^{55} +6.00000 q^{57} +(-2.12132 - 1.22474i) q^{59} +(3.67423 + 6.36396i) q^{61} +(3.00000 + 5.19615i) q^{65} +(-1.00000 + 1.73205i) q^{67} +9.79796 q^{69} -10.0000i q^{71} +(-12.7279 - 7.34847i) q^{73} +(-2.12132 + 1.22474i) q^{75} +(5.19615 - 3.00000i) q^{79} +(4.50000 - 7.79423i) q^{81} -2.44949i q^{83} -12.0000i q^{85} +(-4.89898 + 8.48528i) q^{87} +(-12.7279 + 7.34847i) q^{89} +(10.3923 - 6.00000i) q^{93} +(-5.19615 - 3.00000i) q^{95} -4.89898i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{9} + 8 q^{11} - 4 q^{25} + 48 q^{43} + 48 q^{51} + 48 q^{57} + 24 q^{65} - 8 q^{67} + 36 q^{81} + 48 q^{99}+O(q^{100}) \) 8 * q + 12 * q^9 + 8 * q^11 - 4 * q^25 + 48 * q^43 + 48 * q^51 + 48 * q^57 + 24 * q^65 - 8 * q^67 + 36 * q^81 + 48 * q^99

Introduction

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