LMFDB - Embedded newform 539.2.s.a.215.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,2,Mod(19,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([25, 9]))

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

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

chi := DirichletCharacter("539.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.s (of order $30$, degree $8$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.30393666895$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{30})$
Coefficient field:	16.0.9234096523681640625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + \cdots + 256 $ x^16 - x^15 + 2x^14 - 5x^13 + 5x^12 + x^11 + 6x^10 + 5x^9 - 21x^8 + 10x^7 + 24x^6 + 8x^5 + 80x^4 - 160x^3 + 128x^2 - 128*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{U}(1)[D_{30}]$

Embedding invariants

Embedding label			215.2
Root			$1.26336 - 0.635539i$ of defining polynomial
Character	$\chi$	$=$	539.215
Dual form			539.2.s.a.178.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.567234 - 2.66862i) q^{2} +(-4.97271 - 2.21399i) q^{4} +(-5.52176 + 7.60006i) q^{8} +(-2.93444 - 0.623735i) q^{9} +O(q^{10})$	$q+(0.567234 - 2.66862i) q^{2} +(-4.97271 - 2.21399i) q^{4} +(-5.52176 + 7.60006i) q^{8} +(-2.93444 - 0.623735i) q^{9} +(-2.50638 - 2.17211i) q^{11} +(9.86500 + 10.9562i) q^{16} +(-3.32903 + 7.47712i) q^{18} +(-7.21825 + 5.45649i) q^{22} +(-1.68093 + 2.91146i) q^{23} +(-0.522642 + 4.97261i) q^{25} +(-3.04028 - 4.18459i) q^{29} +(18.5625 - 10.7171i) q^{32} +(13.2112 + 9.59849i) q^{36} +(-1.15761 - 11.0140i) q^{37} -13.0478i q^{43} +(7.65445 + 16.3504i) q^{44} +(6.81612 + 6.13726i) q^{46} +(12.9736 + 4.21537i) q^{50} +(-4.66710 + 5.18334i) q^{53} +(-12.8917 + 5.73974i) q^{58} +(-8.95886 - 27.5725i) q^{64} +(-6.93074 - 12.0044i) q^{67} +(0.0272696 - 0.0839271i) q^{71} +(20.9437 - 18.8578i) q^{72} +(-30.0488 - 3.15825i) q^{74} +(0.561977 - 2.64389i) q^{79} +(8.22191 + 3.66063i) q^{81} +(-34.8198 - 7.40117i) q^{86} +(30.3478 - 7.05472i) q^{88} +(14.8048 - 10.7563i) q^{92} +(6.00000 + 7.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 10 q^{4} - 20 q^{8} + 6 q^{9}+O(q^{10}) $ 16 * q - 10 * q^4 - 20 * q^8 + 6 * q^9	$ 16 q - 10 q^{4} - 20 q^{8} + 6 q^{9} + 4 q^{11} - 8 q^{16} + 15 q^{18} - 28 q^{22} - 16 q^{23} + 10 q^{25} + 60 q^{36} + 18 q^{37} - 25 q^{44} - 15 q^{46} - 30 q^{53} - 19 q^{58} - 68 q^{64} - 8 q^{67} - 96 q^{71} + 75 q^{72} + 40 q^{79} + 18 q^{81} - 23 q^{86} + 8 q^{88} + 50 q^{92} + 96 q^{99}+O(q^{100}) $ 16 * q - 10 * q^4 - 20 * q^8 + 6 * q^9 + 4 * q^11 - 8 * q^16 + 15 * q^18 - 28 * q^22 - 16 * q^23 + 10 * q^25 + 60 * q^36 + 18 * q^37 - 25 * q^44 - 15 * q^46 - 30 * q^53 - 19 * q^58 - 68 * q^64 - 8 * q^67 - 96 * q^71 + 75 * q^72 + 40 * q^79 + 18 * q^81 - 23 * q^86 + 8 * q^88 + 50 * q^92 + 96 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$.

$n$	$199$	$442$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{9}{10}\right)$

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 $

$2$	0.567234	−	2.66862i	0.401095	−	1.88700i	−0.0574802	−	0.998347i	$-0.518307\pi$
$2$	0.567234	−	2.66862i	0.401095	−	1.88700i	0.458575	−	0.888656i	$-0.348360\pi$
$3$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$3$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$4$	−4.97271	−	2.21399i	−2.48636	−	1.10700i
$5$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$5$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−5.52176	+	7.60006i	−1.95224	+	2.68703i
$9$	−2.93444	−	0.623735i	−0.978148	−	0.207912i
$10$		0			0
$11$	−2.50638	−	2.17211i	−0.755701	−	0.654916i
$11$	−2.50638	−	2.17211i	−0.755701	−	0.654916i
$12$		0			0
$13$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$13$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$14$		0			0
$15$		0			0
$16$	9.86500	+	10.9562i	2.46625	+	2.73905i
$17$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$17$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$18$	−3.32903	+	7.47712i	−0.784660	+	1.76237i
$19$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$19$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$20$		0			0
$21$		0			0
$22$	−7.21825	+	5.45649i	−1.53894	+	1.16333i
$23$	−1.68093	+	2.91146i	−0.350499	+	0.607082i	−0.986337	−	0.164740i	$-0.947321\pi$
$23$	−1.68093	+	2.91146i	−0.350499	+	0.607082i	0.635838	+	0.771823i	$0.280655\pi$
$24$		0			0
$25$	−0.522642	+	4.97261i	−0.104528	+	0.994522i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.04028	−	4.18459i	−0.564567	−	0.777059i	0.427331	−	0.904095i	$-0.359454\pi$
$29$	−3.04028	−	4.18459i	−0.564567	−	0.777059i	−0.991898	+	0.127036i	$0.959454\pi$
$30$		0			0
$31$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$31$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$32$	18.5625	−	10.7171i	3.28142	−	1.89453i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	13.2112	+	9.59849i	2.20187	+	1.59975i
$37$	−1.15761	−	11.0140i	−0.190311	−	1.81069i	−0.506772	−	0.862080i	$-0.669162\pi$
$37$	−1.15761	−	11.0140i	−0.190311	−	1.81069i	0.316462	−	0.948605i	$-0.397505\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$41$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$42$		0			0
$43$		−	13.0478i		−	1.98978i	−0.100978	−	0.994889i	$-0.532197\pi$
$43$		−	13.0478i		−	1.98978i	0.100978	−	0.994889i	$-0.467803\pi$
$44$	7.65445	+	16.3504i	1.15395	+	2.46491i
$45$		0			0
$46$	6.81612	+	6.13726i	1.00498	+	0.904890i
$47$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$47$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$48$		0			0
$49$		0			0
$50$	12.9736	+	4.21537i	1.83474	+	0.596143i
$51$		0			0
$52$		0			0
$53$	−4.66710	+	5.18334i	−0.641075	+	0.711986i	−0.972867	−	0.231367i	$-0.925680\pi$
$53$	−4.66710	+	5.18334i	−0.641075	+	0.711986i	0.331792	+	0.943353i	$0.392347\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−12.8917	+	5.73974i	−1.69276	+	0.753664i
$59$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$59$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$60$		0			0
$61$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$61$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$62$		0			0
$63$		0			0
$64$	−8.95886	−	27.5725i	−1.11986	−	3.44657i
$65$		0			0
$66$		0			0
$67$	−6.93074	−	12.0044i	−0.846725	−	1.46657i	−0.884115	−	0.467270i	$-0.845238\pi$
$67$	−6.93074	−	12.0044i	−0.846725	−	1.46657i	0.0373896	−	0.999301i	$-0.488096\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.0272696	−	0.0839271i	0.00323630	−	0.00996031i	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	0.0272696	−	0.0839271i	0.00323630	−	0.00996031i	0.952662	+	0.304033i	$0.0983332\pi$
$72$	20.9437	−	18.8578i	2.46824	−	2.22241i
$73$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$73$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$74$	−30.0488	−	3.15825i	−3.49310	−	0.367140i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.561977	−	2.64389i	0.0632273	−	0.297461i	−0.935160	−	0.354225i	$-0.884745\pi$
$79$	0.561977	−	2.64389i	0.0632273	−	0.297461i	0.998388	+	0.0567635i	$0.0180781\pi$
$80$		0			0
$81$	8.22191	+	3.66063i	0.913545	+	0.406737i
$82$		0			0
$83$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$83$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$84$		0			0
$85$		0			0
$86$	−34.8198	−	7.40117i	−3.75471	−	0.798089i
$87$		0			0
$88$	30.3478	−	7.05472i	3.23509	−	0.752036i
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$	14.8048	−	10.7563i	1.54350	−	1.12142i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$97$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$98$		0			0
$99$	6.00000	+	7.93725i	0.603023	+	0.797724i
$100$	13.6083	−	23.5702i	1.36083	−	2.35702i
$101$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$101$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$102$		0			0
$103$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$103$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$104$		0			0
$105$		0			0
$106$	11.1850	+	15.3949i	1.08639	+	1.49528i
$107$	3.04027	+	6.82856i	0.293914	+	0.660142i	0.998789	−	0.0491893i	$-0.0156638\pi$
$107$	3.04027	+	6.82856i	0.293914	+	0.660142i	−0.704875	+	0.709331i	$0.748997\pi$
$108$		0			0
$109$	−1.74790	+	1.00915i	−0.167419	+	0.0966592i	−0.581368	−	0.813641i	$-0.697482\pi$
$109$	−1.74790	+	1.00915i	−0.167419	+	0.0966592i	0.413949	+	0.910300i	$0.364149\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−16.7856	−	12.1954i	−1.57905	−	1.14725i	−0.917769	−	0.397114i	$-0.870012\pi$
$113$	−16.7856	−	12.1954i	−1.57905	−	1.14725i	−0.661285	−	0.750135i	$-0.729988\pi$
$114$		0			0
$115$		0			0
$116$	5.85380	+	27.5399i	0.543511	+	2.55702i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.56386	+	10.8883i	0.142169	+	0.989842i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.26991	−	1.06246i	−0.290157	−	0.0942778i	0.160322	−	0.987065i	$-0.448747\pi$
$127$	−3.26991	−	1.06246i	−0.290157	−	0.0942778i	−0.450479	+	0.892787i	$0.648747\pi$
$128$	−36.0291	+	3.78681i	−3.18456	+	0.334710i
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$	−35.9666	+	11.6863i	−3.10704	+	1.00954i
$135$		0			0
$136$		0			0
$137$	4.25582	−	0.904603i	0.363599	−	0.0772854i	−0.0224894	−	0.999747i	$-0.507159\pi$
$137$	4.25582	−	0.904603i	0.363599	−	0.0772854i	0.386089	+	0.922462i	$0.373826\pi$
$138$		0			0
$139$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$139$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$140$		0			0
$141$		0			0
$142$	−0.208502	−	0.120378i	−0.0174971	−	0.0101019i
$143$		0			0
$144$	−22.1145	−	38.3034i	−1.84288	−	3.19195i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−18.6284	+	57.3322i	−1.53124	+	4.71268i
$149$	−7.86471	+	7.08141i	−0.644302	+	0.580132i	−0.925141	−	0.379623i	$-0.876054\pi$
$149$	−7.86471	+	7.08141i	−0.644302	+	0.580132i	0.280840	+	0.959755i	$0.409387\pi$
$150$		0			0
$151$	18.2870	+	1.92204i	1.48818	+	0.156414i	0.813442	−	0.581646i	$-0.197591\pi$
$151$	18.2870	+	1.92204i	1.48818	+	0.156414i	0.674735	+	0.738060i	$0.264258\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$157$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$158$	−6.73679	−	2.99941i	−0.535950	−	0.238620i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	14.4326	−	19.8648i	1.13393	−	1.56072i
$163$	24.9537	+	5.30406i	1.95452	+	0.415446i	0.986112	+	0.166082i	$0.0531115\pi$
$163$	24.9537	+	5.30406i	1.95452	+	0.415446i	0.968410	+	0.249365i	$0.0802218\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$167$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$168$		0			0
$169$	10.5172	−	7.64121i	0.809017	−	0.587785i
$170$		0			0
$171$		0			0
$172$	−28.8878	+	64.8832i	−2.20268	+	4.94729i
$173$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$173$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$174$		0			0
$175$		0			0
$176$	−0.927341	−	48.8882i	−0.0699009	−	3.68509i
$177$		0			0
$178$		0			0
$179$	1.96382	−	18.6845i	0.146783	−	1.39654i	−0.634768	−	0.772703i	$-0.718904\pi$
$179$	1.96382	−	18.6845i	0.146783	−	1.39654i	0.781551	−	0.623841i	$-0.214429\pi$
$180$		0			0
$181$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$181$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$182$		0			0
$183$		0			0
$184$	−12.8456	−	28.8516i	−0.946988	−	2.12697i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.88862	−	27.4833i	−0.209013	−	1.98862i	−0.124921	−	0.992167i	$-0.539868\pi$
$191$	−2.88862	−	27.4833i	−0.209013	−	1.98862i	−0.0840922	−	0.996458i	$-0.526799\pi$
$192$		0			0
$193$	−2.19936	−	10.3472i	−0.158314	−	0.744808i	−0.983639	−	0.180150i	$-0.942342\pi$
$193$	−2.19936	−	10.3472i	−0.158314	−	0.744808i	0.825325	−	0.564657i	$-0.190992\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			23.8442i			1.69883i	0.527724	+	0.849416i	$0.323046\pi$
$197$			23.8442i			1.69883i	−0.527724	+	0.849416i	$0.676954\pi$
$198$	24.5850	−	11.5095i	1.74718	−	0.817942i
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−34.9062	−	31.4297i	−2.46824	−	2.22241i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	6.74859	−	7.49507i	0.469059	−	0.520943i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	13.6488	−	4.43475i	0.939619	−	0.305301i	0.201129	−	0.979565i	$-0.435539\pi$
$211$	13.6488	−	4.43475i	0.939619	−	0.305301i	0.738490	+	0.674264i	$0.235539\pi$
$212$	34.6840	−	15.4423i	2.38211	−	1.06058i
$213$		0			0
$214$	19.9474	−	4.23995i	1.36358	−	0.289837i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.70158	+	5.23692i	0.115245	+	0.354689i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$		0			0
$225$	4.63525	−	14.2658i	0.309017	−	0.951057i
$226$	−42.0664	+	37.8767i	−2.79821	+	2.51952i
$227$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$227$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$228$		0			0
$229$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$229$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$230$		0			0
$231$		0			0
$232$	48.5909			3.19015
$233$	−4.40066	+	20.7035i	−0.288297	+	1.35633i	0.560738	+	0.827993i	$0.310518\pi$
$233$	−4.40066	+	20.7035i	−0.288297	+	1.35633i	−0.849035	+	0.528337i	$0.822816\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.7499	+	18.9251i	−0.889407	+	1.22416i	0.0843185	+	0.996439i	$0.473129\pi$
$239$	−13.7499	+	18.9251i	−0.889407	+	1.22416i	−0.973726	+	0.227725i	$0.926871\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$	29.9438	+	2.00284i	1.92486	+	0.128747i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$		0			0
$253$	10.5371	−	3.64605i	0.662461	−	0.229225i
$254$	−4.69010	+	8.12349i	−0.294283	+	0.509713i
$255$		0			0
$256$	−4.27048	+	40.6309i	−0.266905	+	2.53943i
$257$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$257$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.31146	+	14.1758i	0.390670	+	0.877459i
$262$		0			0
$263$	24.9404	−	14.3993i	1.53789	−	0.887900i	0.538926	−	0.842353i	$-0.318830\pi$
$263$	24.9404	−	14.3993i	1.53789	−	0.887900i	0.998962	−	0.0455472i	$-0.0145031\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	7.88692	+	75.0390i	0.481770	+	4.58374i
$269$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$269$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$270$		0			0
$271$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$271$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$272$		0			0
$273$		0			0
$274$		−	11.8703i		−	0.717112i
$275$	12.1110	−	11.3280i	0.730321	−	0.683104i
$276$		0			0
$277$	14.7199	+	13.2539i	0.884436	+	0.796350i	0.979984	−	0.199075i	$-0.0637938\pi$
$277$	14.7199	+	13.2539i	0.884436	+	0.796350i	−0.0955484	+	0.995425i	$0.530460\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−29.7378	−	9.66238i	−1.77401	−	0.576409i	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−29.7378	−	9.66238i	−1.77401	−	0.576409i	−0.998491	+	0.0549198i	$0.982510\pi$
$282$		0			0
$283$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$283$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$284$	−0.321418	+	0.356970i	−0.0190726	+	0.0211823i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−61.1552	+	19.8705i	−3.60360	+	1.17088i
$289$	−15.5303	+	6.91452i	−0.913545	+	0.406737i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$293$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$294$		0			0
$295$		0			0
$296$	90.0989	+	52.0186i	5.23689	+	3.02352i
$297$		0			0
$298$	14.4365	+	25.0048i	0.836284	+	1.44849i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	15.5022	−	47.7110i	0.892053	−	2.74546i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$311$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$312$		0			0
$313$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$313$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$314$		0			0
$315$		0			0
$316$	−8.64811	+	11.9031i	−0.486494	+	0.669602i
$317$	20.1221	+	4.27708i	1.13017	+	0.240225i	0.734791	−	0.678294i	$-0.237280\pi$
$317$	20.1221	+	4.27708i	1.13017	+	0.240225i	0.395378	+	0.918519i	$0.370614\pi$
$318$		0			0
$319$	−1.46930	+	17.0920i	−0.0822651	+	0.956969i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−32.7806	−	36.4065i	−1.82114	−	2.02258i
$325$		0			0
$326$	28.3091	−	63.5833i	1.56790	−	3.52155i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	16.1174	−	27.9162i	0.885895	−	1.53441i	0.0412097	−	0.999151i	$-0.486879\pi$
$331$	16.1174	−	27.9162i	0.885895	−	1.53441i	0.844685	−	0.535264i	$-0.179788\pi$
$332$		0			0
$333$	−3.47284	+	33.0419i	−0.190311	+	1.81069i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	20.4298	+	28.1192i	1.11288	+	1.53175i	0.817102	+	0.576493i	$0.195579\pi$
$337$	20.4298	+	28.1192i	1.11288	+	1.53175i	0.295779	+	0.955256i	$0.404421\pi$
$338$	−14.4258	−	32.4009i	−0.784660	−	1.76237i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	99.1643	+	72.0471i	5.34658	+	3.88452i
$345$		0			0
$346$		0			0
$347$	−1.58885	−	7.47493i	−0.0852937	−	0.401275i	0.914702	−	0.404128i	$-0.132425\pi$
$347$	−1.58885	−	7.47493i	−0.0852937	−	0.401275i	−0.999996	+	0.00285322i	$0.999092\pi$
$348$		0			0
$349$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$349$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$350$		0			0
$351$		0			0
$352$	−69.8033	−	13.4588i	−3.72053	−	0.717357i
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−48.7479	−	15.8392i	−2.57641	−	0.837126i
$359$	25.1257	−	2.64082i	1.32609	−	0.139377i	0.585134	−	0.810936i	$-0.301042\pi$
$359$	25.1257	−	2.64082i	1.32609	−	0.139377i	0.740951	+	0.671559i	$0.234375\pi$
$360$		0			0
$361$	−12.7135	+	14.1198i	−0.669131	+	0.743145i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$367$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$368$	−48.4810	+	10.3049i	−2.52724	+	0.537182i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−27.4955	−	15.8745i	−1.42366	−	0.821951i	−0.427051	−	0.904227i	$-0.640448\pi$
$373$	−27.4955	−	15.8745i	−1.42366	−	0.821951i	−0.996610	+	0.0822766i	$0.973781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	3.72788	−	11.4732i	0.191488	−	0.589340i	−0.808511	−	0.588481i	$-0.799726\pi$
$379$	3.72788	−	11.4732i	0.191488	−	0.589340i	1.00000		0.000859657i	$-0.000273637\pi$
$380$		0			0
$381$		0			0
$382$	−74.9812	−	7.88084i	−3.83637	−	0.403219i
$383$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$383$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$384$		0			0
$385$		0			0
$386$	−28.8603			−1.46895
$387$	−8.13840	+	38.2881i	−0.413698	+	1.94630i
$388$		0			0
$389$	−22.4021	−	9.97404i	−1.13583	−	0.505704i	−0.249323	−	0.968420i	$-0.580208\pi$
$389$	−22.4021	−	9.97404i	−1.13583	−	0.505704i	−0.886506	+	0.462716i	$0.846875\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	63.6313	+	13.5253i	3.20570	+	0.681393i
$395$		0			0
$396$	−12.2632	−	52.7536i	−0.616251	−	2.65097i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	−59.6367	+	43.3286i	−2.98184	+	2.16643i
$401$	−26.7302	−	29.6869i	−1.33484	−	1.48249i	−0.720509	−	0.693446i	$-0.756092\pi$
$401$	−26.7302	−	29.6869i	−1.33484	−	1.48249i	−0.614333	−	0.789047i	$-0.710575\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.0222	+	30.1196i	−1.04203	+	1.49297i
$408$		0			0
$409$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$409$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−16.1735	−	22.2609i	−0.794884	−	1.09406i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.91967	+	1.39472i	0.0935588	+	0.0679745i	0.633581	−	0.773676i	$-0.281584\pi$
$421$	1.91967	+	1.39472i	0.0935588	+	0.0679745i	−0.540022	+	0.841651i	$0.681584\pi$
$422$	−4.09265	−	38.9390i	−0.199227	−	1.89552i
$423$		0			0
$424$	−13.6230	−	64.0914i	−0.661593	−	3.11255i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		−	40.6876i		−	1.96671i
$429$		0			0
$430$		0			0
$431$	−29.8846	−	26.9082i	−1.43949	−	1.29612i	−0.886334	−	0.463046i	$-0.846756\pi$
$431$	−29.8846	−	26.9082i	−1.43949	−	1.29612i	−0.553157	−	0.833077i	$-0.686577\pi$
$432$		0			0
$433$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$433$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$	10.9261	−	1.14838i	0.523264	−	0.0549972i
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−37.8281	+	16.8421i	−1.79727	+	0.800194i	−0.825441	+	0.564489i	$0.809073\pi$
$443$	−37.8281	+	16.8421i	−1.79727	+	0.800194i	−0.971825	+	0.235705i	$0.924260\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8.18900	+	25.2032i	0.386463	+	1.18941i	0.935413	+	0.353556i	$0.115028\pi$
$449$	8.18900	+	25.2032i	0.386463	+	1.18941i	−0.548950	+	0.835855i	$0.684972\pi$
$450$	−35.4409	−	20.4618i	−1.67070	−	0.964580i
$451$		0			0
$452$	56.4692	+	97.8075i	2.65609	+	4.60048i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−22.8299	+	20.5561i	−1.06794	+	0.961574i	−0.999349	−	0.0360712i	$-0.988516\pi$
$457$	−22.8299	+	20.5561i	−1.06794	+	0.961574i	−0.0685868	+	0.997645i	$0.521849\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	27.4582			1.27609			0.638046	−	0.769998i	$-0.279743\pi$
$463$	27.4582			1.27609			0.638046	+	0.769998i	$0.279743\pi$
$464$	15.8548	−	74.5909i	0.736040	−	3.46280i
$465$		0			0
$466$	52.7536	+	23.4874i	2.44376	+	1.08803i
$467$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$467$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−28.3414	+	32.7028i	−1.30314	+	1.50368i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	16.9284	−	12.2992i	0.775096	−	0.563140i
$478$	42.7046	+	47.4283i	1.95326	+	2.16932i
$479$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$479$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	16.3299	−	57.6066i	0.742270	−	2.61848i
$485$		0			0
$486$		0			0
$487$	2.90706	−	27.6589i	0.131732	−	1.25334i	−0.706376	−	0.707837i	$-0.749671\pi$
$487$	2.90706	−	27.6589i	0.131732	−	1.25334i	0.838107	−	0.545506i	$-0.183662\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.11027	+	4.28092i	0.140364	+	0.193195i	0.873412	−	0.486983i	$-0.161903\pi$
$491$	3.11027	+	4.28092i	0.140364	+	0.193195i	−0.733047	+	0.680178i	$0.761903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.46737	−	13.9611i	−0.0656884	−	0.624984i	−0.976996	−	0.213257i	$-0.931593\pi$
$499$	−1.46737	−	13.9611i	−0.0656884	−	0.624984i	0.911308	−	0.411726i	$-0.135074\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$503$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$504$		0			0
$505$		0			0
$506$	−3.75295	−	30.1877i	−0.166839	−	1.34201i
$507$		0			0
$508$	13.9080	+	12.5228i	0.617069	+	0.555611i
$509$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$509$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$510$		0			0
$511$		0			0
$512$	37.0972	+	12.0536i	1.63948	+	0.532700i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$521$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$522$	41.4099	−	8.80195i	1.81246	−	0.385251i
$523$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$523$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$524$		0			0
$525$		0			0
$526$	−24.2794	−	74.7242i	−1.05863	−	3.25813i
$527$		0			0
$528$		0			0
$529$	5.84892	+	10.1306i	0.254301	+	0.440462i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	129.504	+	13.6114i	5.59372	+	0.587924i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9.49537	−	44.6722i	0.408238	−	1.92061i	0.0181362	−	0.999836i	$-0.494227\pi$
$541$	9.49537	−	44.6722i	0.408238	−	1.92061i	0.390101	−	0.920772i	$-0.372440\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−9.33080	+	12.8427i	−0.398956	+	0.549116i	−0.960482	−	0.278343i	$-0.910215\pi$
$547$	−9.33080	+	12.8427i	−0.398956	+	0.549116i	0.561525	+	0.827460i	$0.310215\pi$
$548$	−23.1658	−	4.92403i	−0.989592	−	0.210344i
$549$		0			0
$550$	−23.3604	−	38.7453i	−0.996091	−	1.65211i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	43.7193	−	31.7639i	1.85746	−	1.34952i
$555$		0			0
$556$		0			0
$557$	−19.1243	+	42.9539i	−0.810323	+	1.82002i	−0.324847	+	0.945767i	$0.605313\pi$
$557$	−19.1243	+	42.9539i	−0.810323	+	1.82002i	−0.485476	+	0.874250i	$0.661354\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−42.6535	+	73.8781i	−1.79923	+	3.11636i
$563$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$563$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	0.487274	+	0.670676i	0.0204456	+	0.0281409i
$569$	17.2180	+	38.6722i	0.721815	+	1.62122i	0.782184	+	0.623048i	$0.214106\pi$
$569$	17.2180	+	38.6722i	0.721815	+	1.62122i	−0.0603683	+	0.998176i	$0.519228\pi$
$570$		0			0
$571$	−31.3303	+	18.0885i	−1.31113	+	0.756982i	−0.982284	−	0.187401i	$-0.939994\pi$
$571$	−31.3303	+	18.0885i	−1.31113	+	0.756982i	−0.328848	+	0.944383i	$0.606660\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−13.5990	−	9.88028i	−0.567119	−	0.412036i
$576$	9.09131	+	86.4980i	0.378805	+	3.60408i
$577$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$577$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$578$	9.64297	+	45.3666i	0.401095	+	1.88700i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	22.9563	−	2.85394i	0.950752	−	0.118198i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$587$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	109.251	−	121.336i	4.49020	−	4.98687i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$	54.7871	−	17.8014i	2.24417	−	0.729174i
$597$		0			0
$598$		0			0
$599$	−46.6189	+	9.90916i	−1.90480	+	0.404877i	−0.999789	−	0.0205175i	$-0.993469\pi$
$599$	−46.6189	+	9.90916i	−1.90480	+	0.404877i	−0.905008	+	0.425395i	$0.860135\pi$
$600$		0			0
$601$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$601$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$602$		0			0
$603$	12.8503	+	39.5492i	0.523305	+	1.61057i
$604$	−86.6807	−	50.0451i	−3.52699	−	2.03631i
$605$		0			0
$606$		0			0
$607$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$607$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	26.1849	+	2.75215i	1.05760	+	0.111158i	0.617327	−	0.786706i	$-0.288215\pi$
$613$	26.1849	+	2.75215i	1.05760	+	0.111158i	0.440272	+	0.897865i	$0.354882\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−48.2946			−1.94427			−0.972133	−	0.234428i	$-0.924678\pi$
$617$	−48.2946			−1.94427			−0.972133	+	0.234428i	$0.924678\pi$
$618$		0			0
$619$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$619$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.4537	−	5.19779i	−0.978148	−	0.207912i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	33.1185	−	24.0620i	1.31843	−	0.957892i	0.318475	−	0.947931i	$-0.396829\pi$
$631$	33.1185	−	24.0620i	1.31843	−	0.957892i	0.999950	−	0.00996082i	$-0.00317068\pi$
$632$	16.9906	+	18.8700i	0.675851	+	0.750609i
$633$		0			0
$634$	22.8278	−	51.2722i	0.906609	−	2.03628i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	44.7787	+	13.6162i	1.77281	+	0.539070i
$639$	−0.132369	+	0.229270i	−0.00523645	+	0.00906979i
$640$		0			0
$641$	−3.59001	+	34.1567i	−0.141797	+	1.34911i	0.659889	+	0.751363i	$0.270603\pi$
$641$	−3.59001	+	34.1567i	−0.141797	+	1.34911i	−0.801686	+	0.597746i	$0.796063\pi$
$642$		0			0
$643$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$643$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$647$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$648$	−73.2204	+	42.2738i	−2.87637	+	1.66067i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−112.344	−	81.6228i	−4.39974	−	3.19660i
$653$	−4.87849	−	46.4157i	−0.190910	−	1.81639i	−0.500743	−	0.865596i	$-0.666940\pi$
$653$	−4.87849	−	46.4157i	−0.190910	−	1.81639i	0.309833	−	0.950791i	$-0.399727\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	26.4575i		−	1.03064i	−0.856998	−	0.515319i	$-0.827673\pi$
$659$		−	26.4575i		−	1.03064i	0.856998	−	0.515319i	$-0.172327\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$	−65.3556	−	58.8464i	−2.54012	−	2.28713i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	86.2065	+	28.0102i	3.34043	+	1.08537i
$667$	17.2938	−	1.81765i	0.669619	−	0.0703798i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−39.5763	+	12.8591i	−1.52556	+	0.495683i	−0.947348	−	0.320207i	$-0.896248\pi$
$673$	−39.5763	+	12.8591i	−1.52556	+	0.495683i	−0.578208	+	0.815890i	$0.696248\pi$
$674$	86.6280	−	38.5693i	3.33679	−	1.48563i
$675$		0			0
$676$	−69.2167	+	14.7125i	−2.66218	+	0.565864i
$677$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$677$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.4793	−	33.7392i	−0.745355	−	1.29099i	−0.950029	−	0.312162i	$-0.898947\pi$
$683$	−19.4793	−	33.7392i	−0.745355	−	1.29099i	0.204674	−	0.978830i	$-0.434387\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	142.955	−	128.717i	5.45009	−	4.90729i
$689$		0			0
$690$		0			0
$691$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$691$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$692$		0			0
$693$		0			0
$694$	−20.8490			−0.791418
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	24.4716	−	33.6823i	0.924280	−	1.27216i	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	24.4716	−	33.6823i	0.924280	−	1.27216i	0.962049	−	0.272876i	$-0.0879747\pi$
$702$		0			0
$703$		0			0
$704$	−37.4364	+	88.5669i	−1.41094	+	3.33799i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−32.4335	−	36.0210i	−1.21806	−	1.35280i	−0.916845	−	0.399244i	$-0.869273\pi$
$709$	−32.4335	−	36.0210i	−1.21806	−	1.35280i	−0.301220	−	0.953555i	$-0.597394\pi$
$710$		0			0
$711$	−3.29818	+	7.40783i	−0.123691	+	0.277815i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−51.1328	+	88.5647i	−1.91092	+	3.30982i
$717$		0			0
$718$	7.20480	−	68.5491i	0.268881	−	2.55823i
$719$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$719$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$720$		0			0
$721$		0			0
$722$	30.4688	+	41.9367i	1.13393	+	1.56072i
$723$		0			0
$724$		0			0
$725$	22.3973	−	12.9311i	0.831816	−	0.480249i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−21.8435	−	15.8702i	−0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$733$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$734$		0			0
$735$		0			0
$736$			72.0587i			2.65612i
$737$	−8.70384	+	45.1419i	−0.320610	+	1.66282i
$738$		0			0
$739$	33.1067	+	29.8094i	1.21785	+	1.09656i	0.992496	+	0.122279i	$0.0390204\pi$
$739$	33.1067	+	29.8094i	1.21785	+	1.09656i	0.225354	+	0.974277i	$0.427646\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	47.0663	+	15.2928i	1.72669	+	0.561037i	0.992965	−	0.118405i	$-0.0377783\pi$
$743$	47.0663	+	15.2928i	1.72669	+	0.561037i	0.733729	+	0.679442i	$0.237778\pi$
$744$		0			0
$745$		0			0
$746$	−57.9594	+	64.3705i	−2.12205	+	2.35677i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−49.6824	+	22.1200i	−1.81294	+	0.807171i	−0.855938	+	0.517079i	$0.827019\pi$
$751$	−49.6824	+	22.1200i	−1.81294	+	0.807171i	−0.956999	+	0.290092i	$0.906314\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.04627	+	6.29778i	0.0743731	+	0.228897i	0.981332	−	0.192323i	$-0.0616021\pi$
$757$	2.04627	+	6.29778i	0.0743731	+	0.228897i	−0.906959	+	0.421220i	$0.861602\pi$
$758$	−28.5032	−	16.4563i	−1.03528	−	0.597720i
$759$		0			0
$760$		0			0
$761$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$761$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$762$		0			0
$763$		0			0
$764$	−46.4837	+	143.062i	−1.68172	+	5.17580i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	−11.9718	+	56.3230i	−0.430875	+	2.02711i
$773$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$773$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$774$	97.5603	+	43.4366i	3.50673	+	1.56130i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−39.3242	+	54.1251i	−1.40984	+	1.94048i
$779$		0			0
$780$		0			0
$781$	−0.250647	+	0.151120i	−0.00896885	+	0.00540751i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$787$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$788$	52.7910	−	118.571i	1.88060	−	4.22390i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−93.4542	+	1.77269i	−3.32075	+	0.0629899i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$797$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$798$		0			0
$799$		0			0
$800$	43.5902	+	97.9052i	1.54115	+	3.46147i
$801$		0			0
$802$	−94.3854	+	54.4934i	−3.33286	+	1.92423i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11.7643	−	55.3467i	−0.413611	−	1.94589i	−0.307576	−	0.951523i	$-0.599518\pi$
$809$	−11.7643	−	55.3467i	−0.413611	−	1.94589i	−0.106034	−	0.994362i	$-0.533815\pi$
$810$		0			0
$811$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$811$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$812$		0			0
$813$		0			0
$814$	68.4535	+	73.1851i	2.39929	+	2.56514i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	52.6252	−	5.53113i	1.83663	−	0.193038i	0.878186	−	0.478318i	$-0.158754\pi$
$821$	52.6252	−	5.53113i	1.83663	−	0.193038i	0.958444	+	0.285281i	$0.0920868\pi$
$822$		0			0
$823$	36.9234	−	41.0076i	1.28707	−	1.42944i	0.439961	−	0.898017i	$-0.354992\pi$
$823$	36.9234	−	41.0076i	1.28707	−	1.42944i	0.847109	−	0.531419i	$-0.178341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.2276	+	11.4461i	−1.22498	+	0.398022i	−0.848895	−	0.528562i	$-0.822732\pi$
$827$	−35.2276	+	11.4461i	−1.22498	+	0.398022i	−0.376090	+	0.926583i	$0.622732\pi$
$828$	−50.1528	+	22.3295i	−1.74293	+	0.776003i
$829$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$829$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$839$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$840$		0			0
$841$	0.694005	−	2.13593i	0.0239312	−	0.0736527i
$842$	4.81088	−	4.33174i	0.165794	−	0.149282i
$843$		0			0
$844$	−77.6898	−	8.16553i	−2.67419	−	0.281069i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−102.830			−3.53121
$849$		0			0
$850$		0			0
$851$	34.0126	+	15.1434i	1.16594	+	0.519109i
$852$		0			0
$853$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$853$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$854$		0			0
$855$		0			0
$856$	−68.6851	−	14.5995i	−2.34761	−	0.499000i
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$	−88.7595	+	64.4875i	−3.02316	+	2.19645i
$863$	5.35304	+	5.94516i	0.182220	+	0.202376i	0.827334	−	0.561711i	$-0.189857\pi$
$863$	5.35304	+	5.94516i	0.182220	+	0.202376i	−0.645114	+	0.764087i	$0.723190\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.15136	+	5.40592i	−0.242593	+	0.183383i
$870$		0			0
$871$		0			0
$872$	1.98189	−	18.8565i	0.0671154	−	0.638560i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.3320	+	52.4044i	0.787864	+	1.76957i	0.621349	+	0.783534i	$0.286585\pi$
$877$	23.3320	+	52.4044i	0.787864	+	1.76957i	0.166515	+	0.986039i	$0.446749\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	9.70820	+	7.05342i	0.326707	+	0.237367i	0.739032	−	0.673670i	$-0.235283\pi$
$883$	9.70820	+	7.05342i	0.326707	+	0.237367i	−0.412325	+	0.911037i	$0.635283\pi$
$884$		0			0
$885$		0			0
$886$	23.4880	+	110.502i	0.789094	+	3.71240i
$887$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$887$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−12.6559	−	27.0338i	−0.423989	−	0.905667i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	71.9028	−	7.55729i	2.39943	−	0.252190i
$899$		0			0
$900$	−54.6343	+	60.6775i	−1.82114	+	2.02258i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	185.372	−	60.2310i	6.16538	−	2.00325i
$905$		0			0
$906$		0			0
$907$	13.2133	−	2.80858i	0.438741	−	0.0932572i	0.0167548	−	0.999860i	$-0.494667\pi$
$907$	13.2133	−	2.80858i	0.438741	−	0.0932572i	0.421986	+	0.906602i	$0.361333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.94427	−	15.2169i	−0.163811	−	0.504159i	0.835136	−	0.550044i	$-0.185389\pi$
$911$	−4.94427	−	15.2169i	−0.163811	−	0.504159i	−0.998947	+	0.0458855i	$0.985389\pi$
$912$		0			0
$913$		0			0
$914$	41.9066	+	72.5844i	1.38615	+	2.40088i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	25.4189	−	22.8873i	0.838494	−	0.754983i	−0.133235	−	0.991084i	$-0.542536\pi$
$919$	25.4189	−	22.8873i	0.838494	−	0.754983i	0.971728	+	0.236101i	$0.0758698\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	55.3732			1.82066
$926$	15.5752	−	73.2757i	0.511834	−	2.40799i
$927$		0			0
$928$	−101.282	−	45.0936i	−3.32474	−	1.48027i
$929$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$929$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$930$		0			0
$931$		0			0
$932$	67.7206	−	93.2094i	2.21826	−	3.05318i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$937$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$941$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	71.1954	+	94.1826i	2.31476	+	3.06214i
$947$	10.0000	−	17.3205i	0.324956	−	0.562841i	−0.656547	−	0.754285i	$-0.727984\pi$
$947$	10.0000	−	17.3205i	0.324956	−	0.562841i	0.981504	+	0.191444i	$0.0613171\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.97347	−	13.7273i	−0.323072	−	0.444671i	0.616330	−	0.787488i	$-0.288619\pi$
$953$	−9.97347	−	13.7273i	−0.323072	−	0.444671i	−0.939402	+	0.342817i	$0.888619\pi$
$954$	−23.2195	−	52.1519i	−0.751760	−	1.68848i
$955$		0			0
$956$	110.274	−	63.6669i	3.56653	−	2.05914i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−3.24038	−	30.8302i	−0.104528	−	0.994522i
$962$		0			0
$963$	−4.66229	−	21.9343i	−0.150240	−	0.706824i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	62.0397i		−	1.99506i	−0.0702371	−	0.997530i	$-0.522376\pi$
$967$		−	62.0397i		−	1.99506i	0.0702371	−	0.997530i	$-0.477624\pi$
$968$	−91.3867	−	48.2370i	−2.93728	−	1.55040i
$969$		0			0
$970$		0			0
$971$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$971$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$972$		0			0
$973$		0			0
$974$	−72.1621	−	23.4469i	−2.31222	−	0.751287i
$975$		0			0
$976$		0			0
$977$	41.5509	−	46.1470i	1.32933	−	1.47637i	0.578966	−	0.815352i	$-0.303456\pi$
$977$	41.5509	−	46.1470i	1.32933	−	1.47637i	0.750367	−	0.661022i	$-0.229877\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	5.75856	−	1.87107i	0.183857	−	0.0597387i
$982$	13.1884	−	5.87186i	0.420859	−	0.187378i
$983$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$983$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	37.9883	+	21.9326i	1.20796	+	0.697415i
$990$		0			0
$991$	−12.0000	−	20.7846i	−0.381193	−	0.660245i	0.610040	−	0.792370i	$-0.291153\pi$
$991$	−12.0000	−	20.7846i	−0.381193	−	0.660245i	−0.991233	+	0.132125i	$0.957820\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$997$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$998$	−38.0892	−	4.00334i	−1.20569	−	0.126723i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.s.a.215.2		16
7.2	even	3		77.2.l.a.6.1	✓	8
7.3	odd	6	inner	539.2.s.a.325.2		16
7.4	even	3	inner	539.2.s.a.325.2		16
7.5	odd	6		77.2.l.a.6.1	✓	8
7.6	odd	2	CM	539.2.s.a.215.2		16
11.2	odd	10	inner	539.2.s.a.68.2		16
21.2	odd	6		693.2.bu.a.622.2		8
21.5	even	6		693.2.bu.a.622.2		8
77.2	odd	30		77.2.l.a.13.1	yes	8
77.5	odd	30		847.2.l.d.524.2		8
77.9	even	15		847.2.l.c.475.2		8
77.13	even	10	inner	539.2.s.a.68.2		16
77.16	even	15		847.2.l.d.524.2		8
77.19	even	30		847.2.b.b.846.1		8
77.24	even	30	inner	539.2.s.a.178.2		16
77.26	odd	30		847.2.l.a.118.1		8
77.30	odd	30		847.2.b.b.846.1		8
77.37	even	15		847.2.l.a.118.1		8
77.40	even	30		847.2.l.d.118.2		8
77.46	odd	30	inner	539.2.s.a.178.2		16
77.47	odd	30		847.2.b.b.846.8		8
77.51	odd	30		847.2.l.d.118.2		8
77.54	even	6		847.2.l.c.699.2		8
77.58	even	15		847.2.b.b.846.8		8
77.61	even	30		847.2.l.a.524.1		8
77.65	odd	6		847.2.l.c.699.2		8
77.68	even	30		77.2.l.a.13.1	yes	8
77.72	odd	30		847.2.l.a.524.1		8
77.75	odd	30		847.2.l.c.475.2		8
231.2	even	30		693.2.bu.a.244.2		8
231.68	odd	30		693.2.bu.a.244.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.6.1	✓	8	7.2	even	3
77.2.l.a.6.1	✓	8	7.5	odd	6
77.2.l.a.13.1	yes	8	77.2	odd	30
77.2.l.a.13.1	yes	8	77.68	even	30
539.2.s.a.68.2		16	11.2	odd	10	inner
539.2.s.a.68.2		16	77.13	even	10	inner
539.2.s.a.178.2		16	77.24	even	30	inner
539.2.s.a.178.2		16	77.46	odd	30	inner
539.2.s.a.215.2		16	1.1	even	1	trivial
539.2.s.a.215.2		16	7.6	odd	2	CM
539.2.s.a.325.2		16	7.3	odd	6	inner
539.2.s.a.325.2		16	7.4	even	3	inner
693.2.bu.a.244.2		8	231.2	even	30
693.2.bu.a.244.2		8	231.68	odd	30
693.2.bu.a.622.2		8	21.2	odd	6
693.2.bu.a.622.2		8	21.5	even	6
847.2.b.b.846.1		8	77.19	even	30
847.2.b.b.846.1		8	77.30	odd	30
847.2.b.b.846.8		8	77.47	odd	30
847.2.b.b.846.8		8	77.58	even	15
847.2.l.a.118.1		8	77.26	odd	30
847.2.l.a.118.1		8	77.37	even	15
847.2.l.a.524.1		8	77.61	even	30
847.2.l.a.524.1		8	77.72	odd	30
847.2.l.c.475.2		8	77.9	even	15
847.2.l.c.475.2		8	77.75	odd	30
847.2.l.c.699.2		8	77.54	even	6
847.2.l.c.699.2		8	77.65	odd	6
847.2.l.d.118.2		8	77.40	even	30
847.2.l.d.118.2		8	77.51	odd	30
847.2.l.d.524.2		8	77.5	odd	30
847.2.l.d.524.2		8	77.16	even	15

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.s (of order \(30\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.567234 - 2.66862i) q^{2} +(-4.97271 - 2.21399i) q^{4} +(-5.52176 + 7.60006i) q^{8} +(-2.93444 - 0.623735i) q^{9} +O(q^{10})\)	\(q+(0.567234 - 2.66862i) q^{2} +(-4.97271 - 2.21399i) q^{4} +(-5.52176 + 7.60006i) q^{8} +(-2.93444 - 0.623735i) q^{9} +(-2.50638 - 2.17211i) q^{11} +(9.86500 + 10.9562i) q^{16} +(-3.32903 + 7.47712i) q^{18} +(-7.21825 + 5.45649i) q^{22} +(-1.68093 + 2.91146i) q^{23} +(-0.522642 + 4.97261i) q^{25} +(-3.04028 - 4.18459i) q^{29} +(18.5625 - 10.7171i) q^{32} +(13.2112 + 9.59849i) q^{36} +(-1.15761 - 11.0140i) q^{37} -13.0478i q^{43} +(7.65445 + 16.3504i) q^{44} +(6.81612 + 6.13726i) q^{46} +(12.9736 + 4.21537i) q^{50} +(-4.66710 + 5.18334i) q^{53} +(-12.8917 + 5.73974i) q^{58} +(-8.95886 - 27.5725i) q^{64} +(-6.93074 - 12.0044i) q^{67} +(0.0272696 - 0.0839271i) q^{71} +(20.9437 - 18.8578i) q^{72} +(-30.0488 - 3.15825i) q^{74} +(0.561977 - 2.64389i) q^{79} +(8.22191 + 3.66063i) q^{81} +(-34.8198 - 7.40117i) q^{86} +(30.3478 - 7.05472i) q^{88} +(14.8048 - 10.7563i) q^{92} +(6.00000 + 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{4} - 20 q^{8} + 6 q^{9}+O(q^{10}) \) 16 * q - 10 * q^4 - 20 * q^8 + 6 * q^9	\( 16 q - 10 q^{4} - 20 q^{8} + 6 q^{9} + 4 q^{11} - 8 q^{16} + 15 q^{18} - 28 q^{22} - 16 q^{23} + 10 q^{25} + 60 q^{36} + 18 q^{37} - 25 q^{44} - 15 q^{46} - 30 q^{53} - 19 q^{58} - 68 q^{64} - 8 q^{67} - 96 q^{71} + 75 q^{72} + 40 q^{79} + 18 q^{81} - 23 q^{86} + 8 q^{88} + 50 q^{92} + 96 q^{99}+O(q^{100}) \) 16 * q - 10 * q^4 - 20 * q^8 + 6 * q^9 + 4 * q^11 - 8 * q^16 + 15 * q^18 - 28 * q^22 - 16 * q^23 + 10 * q^25 + 60 * q^36 + 18 * q^37 - 25 * q^44 - 15 * q^46 - 30 * q^53 - 19 * q^58 - 68 * q^64 - 8 * q^67 - 96 * q^71 + 75 * q^72 + 40 * q^79 + 18 * q^81 - 23 * q^86 + 8 * q^88 + 50 * q^92 + 96 * q^99

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