LMFDB - Embedded newform 378.2.k.d.215.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(215,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, degree $2$, 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:	$3.01834519640$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			215.2
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	378.215
Dual form			378.2.k.d.269.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.22474 - 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.22474 - 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +(-2.12132 + 1.22474i) q^{10} +(3.67423 - 2.12132i) q^{11} -4.18154i q^{13} +(2.09077 - 1.62132i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.22474 + 2.12132i) q^{17} +(4.24264 + 2.44949i) q^{19} +2.44949 q^{20} -4.24264 q^{22} +(-5.19615 - 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.09077 + 3.62132i) q^{26} +(-2.62132 + 0.358719i) q^{28} -10.2426i q^{29} +(4.86396 - 2.80821i) q^{31} +(0.866025 - 0.500000i) q^{32} -2.44949i q^{34} +(3.97141 + 5.12132i) q^{35} +(1.62132 - 2.80821i) q^{37} +(-2.44949 - 4.24264i) q^{38} +(-2.12132 - 1.22474i) q^{40} +2.44949 q^{41} +7.00000 q^{43} +(3.67423 + 2.12132i) q^{44} +(3.00000 + 5.19615i) q^{46} +(-3.97141 + 6.87868i) q^{47} +(-5.00000 - 4.89898i) q^{49} +1.00000i q^{50} +(3.62132 - 2.09077i) q^{52} +(-2.15232 + 1.24264i) q^{53} -10.3923i q^{55} +(2.44949 + 1.00000i) q^{56} +(-5.12132 + 8.87039i) q^{58} +(-1.22474 - 2.12132i) q^{59} +(-0.621320 - 0.358719i) q^{61} -5.61642 q^{62} -1.00000 q^{64} +(-8.87039 - 5.12132i) q^{65} +(1.74264 + 3.01834i) q^{67} +(-1.22474 + 2.12132i) q^{68} +(-0.878680 - 6.42090i) q^{70} +12.7279i q^{71} +(-13.2426 + 7.64564i) q^{73} +(-2.80821 + 1.62132i) q^{74} +4.89898i q^{76} +(1.52192 + 11.1213i) q^{77} +(-4.62132 + 8.00436i) q^{79} +(1.22474 + 2.12132i) q^{80} +(-2.12132 - 1.22474i) q^{82} +5.49333 q^{83} +6.00000 q^{85} +(-6.06218 - 3.50000i) q^{86} +(-2.12132 - 3.67423i) q^{88} +(-8.87039 + 15.3640i) q^{89} +(10.2426 + 4.18154i) q^{91} -6.00000i q^{92} +(6.87868 - 3.97141i) q^{94} +(10.3923 - 6.00000i) q^{95} +6.63103i q^{97} +(1.88064 + 6.74264i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q + 4 * q^4 - 8 * q^7	$ 8 q + 4 q^{4} - 8 q^{7} - 4 q^{16} - 4 q^{25} - 4 q^{28} - 12 q^{31} - 4 q^{37} + 56 q^{43} + 24 q^{46} - 40 q^{49} + 12 q^{52} - 24 q^{58} + 12 q^{61} - 8 q^{64} - 20 q^{67} - 24 q^{70} - 72 q^{73} - 20 q^{79} + 48 q^{85} + 48 q^{91} + 72 q^{94}+O(q^{100}) $ 8 * q + 4 * q^4 - 8 * q^7 - 4 * q^16 - 4 * q^25 - 4 * q^28 - 12 * q^31 - 4 * q^37 + 56 * q^43 + 24 * q^46 - 40 * q^49 + 12 * q^52 - 24 * q^58 + 12 * q^61 - 8 * q^64 - 20 * q^67 - 24 * q^70 - 72 * q^73 - 20 * q^79 + 48 * q^85 + 48 * q^91 + 72 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$	1.22474	−	2.12132i	0.547723	−	0.948683i	−0.450708	−	0.892672i	$-0.648828\pi$
$5$	1.22474	−	2.12132i	0.547723	−	0.948683i	0.998430	−	0.0560116i	$-0.0178384\pi$
$6$		0			0
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−2.12132	+	1.22474i	−0.670820	+	0.387298i
$11$	3.67423	−	2.12132i	1.10782	−	0.639602i	0.169559	−	0.985520i	$-0.445766\pi$
$11$	3.67423	−	2.12132i	1.10782	−	0.639602i	0.938265	+	0.345918i	$0.112432\pi$
$12$		0			0
$13$		−	4.18154i		−	1.15975i	−0.814705	−	0.579875i	$-0.803101\pi$
$13$		−	4.18154i		−	1.15975i	0.814705	−	0.579875i	$-0.196899\pi$
$14$	2.09077	−	1.62132i	0.558782	−	0.433316i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	1.22474	+	2.12132i	0.297044	+	0.514496i	0.975458	−	0.220184i	$-0.0706658\pi$
$17$	1.22474	+	2.12132i	0.297044	+	0.514496i	−0.678414	+	0.734680i	$0.737332\pi$
$18$		0			0
$19$	4.24264	+	2.44949i	0.973329	+	0.561951i	0.900249	−	0.435375i	$-0.143384\pi$
$19$	4.24264	+	2.44949i	0.973329	+	0.561951i	0.0730792	+	0.997326i	$0.476717\pi$
$20$	2.44949			0.547723
$21$		0			0
$22$	−4.24264			−0.904534
$23$	−5.19615	−	3.00000i	−1.08347	−	0.625543i	−0.151642	−	0.988436i	$-0.548456\pi$
$23$	−5.19615	−	3.00000i	−1.08347	−	0.625543i	−0.931831	+	0.362892i	$0.881789\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	−2.09077	+	3.62132i	−0.410034	+	0.710199i
$27$		0			0
$28$	−2.62132	+	0.358719i	−0.495383	+	0.0677916i
$29$		−	10.2426i		−	1.90201i	−0.309175	−	0.951005i	$-0.600053\pi$
$29$		−	10.2426i		−	1.90201i	0.309175	−	0.951005i	$-0.399947\pi$
$30$		0			0
$31$	4.86396	−	2.80821i	0.873593	−	0.504369i	0.00505256	−	0.999987i	$-0.498392\pi$
$31$	4.86396	−	2.80821i	0.873593	−	0.504369i	0.868541	+	0.495618i	$0.165058\pi$
$32$	0.866025	−	0.500000i	0.153093	−	0.0883883i
$33$		0			0
$34$		−	2.44949i		−	0.420084i
$35$	3.97141	+	5.12132i	0.671290	+	0.865661i
$36$		0			0
$37$	1.62132	−	2.80821i	0.266543	−	0.461667i	−0.701423	−	0.712745i	$-0.747452\pi$
$37$	1.62132	−	2.80821i	0.266543	−	0.461667i	0.967967	+	0.251078i	$0.0807851\pi$
$38$	−2.44949	−	4.24264i	−0.397360	−	0.688247i
$39$		0			0
$40$	−2.12132	−	1.22474i	−0.335410	−	0.193649i
$41$	2.44949			0.382546			0.191273	−	0.981537i	$-0.438738\pi$
$41$	2.44949			0.382546			0.191273	+	0.981537i	$0.438738\pi$
$42$		0			0
$43$	7.00000			1.06749			0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000			1.06749			0.533745	+	0.845645i	$0.320784\pi$
$44$	3.67423	+	2.12132i	0.553912	+	0.319801i
$45$		0			0
$46$	3.00000	+	5.19615i	0.442326	+	0.766131i
$47$	−3.97141	+	6.87868i	−0.579289	+	1.00336i	0.416272	+	0.909240i	$0.363337\pi$
$47$	−3.97141	+	6.87868i	−0.579289	+	1.00336i	−0.995561	+	0.0941183i	$0.969997\pi$
$48$		0			0
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$			1.00000i			0.141421i
$51$		0			0
$52$	3.62132	−	2.09077i	0.502187	−	0.289938i
$53$	−2.15232	+	1.24264i	−0.295643	+	0.170690i	−0.640484	−	0.767971i	$-0.721266\pi$
$53$	−2.15232	+	1.24264i	−0.295643	+	0.170690i	0.344841	+	0.938661i	$0.387933\pi$
$54$		0			0
$55$		−	10.3923i		−	1.40130i
$56$	2.44949	+	1.00000i	0.327327	+	0.133631i
$57$		0			0
$58$	−5.12132	+	8.87039i	−0.672462	+	1.16474i
$59$	−1.22474	−	2.12132i	−0.159448	−	0.276172i	0.775222	−	0.631689i	$-0.217638\pi$
$59$	−1.22474	−	2.12132i	−0.159448	−	0.276172i	−0.934670	+	0.355517i	$0.884305\pi$
$60$		0			0
$61$	−0.621320	−	0.358719i	−0.0795519	−	0.0459293i	0.459696	−	0.888076i	$-0.347958\pi$
$61$	−0.621320	−	0.358719i	−0.0795519	−	0.0459293i	−0.539248	+	0.842147i	$0.681292\pi$
$62$	−5.61642			−0.713286
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−8.87039	−	5.12132i	−1.10024	−	0.635222i
$66$		0			0
$67$	1.74264	+	3.01834i	0.212897	+	0.368749i	0.952620	−	0.304163i	$-0.0983766\pi$
$67$	1.74264	+	3.01834i	0.212897	+	0.368749i	−0.739723	+	0.672912i	$0.765043\pi$
$68$	−1.22474	+	2.12132i	−0.148522	+	0.257248i
$69$		0			0
$70$	−0.878680	−	6.42090i	−0.105022	−	0.767444i
$71$			12.7279i			1.51053i	0.655422	+	0.755263i	$0.272491\pi$
$71$			12.7279i			1.51053i	−0.655422	+	0.755263i	$0.727509\pi$
$72$		0			0
$73$	−13.2426	+	7.64564i	−1.54993	+	0.894855i	−0.551788	+	0.833984i	$0.686054\pi$
$73$	−13.2426	+	7.64564i	−1.54993	+	0.894855i	−0.998146	+	0.0608704i	$0.980612\pi$
$74$	−2.80821	+	1.62132i	−0.326448	+	0.188475i
$75$		0			0
$76$			4.89898i			0.561951i
$77$	1.52192	+	11.1213i	0.173439	+	1.26739i
$78$		0			0
$79$	−4.62132	+	8.00436i	−0.519939	+	0.900561i	0.479792	+	0.877382i	$0.340712\pi$
$79$	−4.62132	+	8.00436i	−0.519939	+	0.900561i	−0.999731	+	0.0231789i	$0.992621\pi$
$80$	1.22474	+	2.12132i	0.136931	+	0.237171i
$81$		0			0
$82$	−2.12132	−	1.22474i	−0.234261	−	0.135250i
$83$	5.49333			0.602971			0.301485	−	0.953471i	$-0.402518\pi$
$83$	5.49333			0.602971			0.301485	+	0.953471i	$0.402518\pi$
$84$		0			0
$85$	6.00000			0.650791
$86$	−6.06218	−	3.50000i	−0.653701	−	0.377415i
$87$		0			0
$88$	−2.12132	−	3.67423i	−0.226134	−	0.391675i
$89$	−8.87039	+	15.3640i	−0.940259	+	1.62858i	−0.175283	+	0.984518i	$0.556084\pi$
$89$	−8.87039	+	15.3640i	−0.940259	+	1.62858i	−0.764976	+	0.644059i	$0.777249\pi$
$90$		0			0
$91$	10.2426	+	4.18154i	1.07372	+	0.438345i
$92$		−	6.00000i		−	0.625543i
$93$		0			0
$94$	6.87868	−	3.97141i	0.709482	−	0.409619i
$95$	10.3923	−	6.00000i	1.06623	−	0.615587i
$96$		0			0
$97$			6.63103i			0.673279i	0.941634	+	0.336640i	$0.109290\pi$
$97$			6.63103i			0.673279i	−0.941634	+	0.336640i	$0.890710\pi$
$98$	1.88064	+	6.74264i	0.189973	+	0.681110i
$99$		0			0
$100$	0.500000	−	0.866025i	0.0500000	−	0.0866025i
$101$	3.67423	+	6.36396i	0.365600	+	0.633238i	0.988872	−	0.148767i	$-0.0475305\pi$
$101$	3.67423	+	6.36396i	0.365600	+	0.633238i	−0.623272	+	0.782005i	$0.714197\pi$
$102$		0			0
$103$	5.37868	+	3.10538i	0.529977	+	0.305982i	0.741007	−	0.671497i	$-0.234348\pi$
$103$	5.37868	+	3.10538i	0.529977	+	0.305982i	−0.211030	+	0.977480i	$0.567682\pi$
$104$	−4.18154			−0.410034
$105$		0			0
$106$	2.48528			0.241392
$107$	−12.5446	−	7.24264i	−1.21273	−	0.700173i	−0.249380	−	0.968406i	$-0.580227\pi$
$107$	−12.5446	−	7.24264i	−1.21273	−	0.700173i	−0.963354	+	0.268233i	$0.913560\pi$
$108$		0			0
$109$	−3.86396	−	6.69258i	−0.370100	−	0.641033i	0.619480	−	0.785012i	$-0.287343\pi$
$109$	−3.86396	−	6.69258i	−0.370100	−	0.641033i	−0.989581	+	0.143980i	$0.954010\pi$
$110$	−5.19615	+	9.00000i	−0.495434	+	0.858116i
$111$		0			0
$112$	−1.62132	−	2.09077i	−0.153200	−	0.197559i
$113$			1.75736i			0.165318i	0.996578	+	0.0826592i	$0.0263413\pi$
$113$			1.75736i			0.165318i	−0.996578	+	0.0826592i	$0.973659\pi$
$114$		0			0
$115$	−12.7279	+	7.34847i	−1.18688	+	0.685248i
$116$	8.87039	−	5.12132i	0.823595	−	0.475503i
$117$		0			0
$118$			2.44949i			0.225494i
$119$	−6.42090	+	0.878680i	−0.588603	+	0.0805484i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$	0.358719	+	0.621320i	0.0324769	+	0.0562517i
$123$		0			0
$124$	4.86396	+	2.80821i	0.436797	+	0.252185i
$125$	9.79796			0.876356
$126$		0			0
$127$	17.7279			1.57310			0.786549	−	0.617527i	$-0.211866\pi$
$127$	17.7279			1.57310			0.786549	+	0.617527i	$0.211866\pi$
$128$	0.866025	+	0.500000i	0.0765466	+	0.0441942i
$129$		0			0
$130$	5.12132	+	8.87039i	0.449170	+	0.777984i
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	−10.2426	+	7.94282i	−0.888150	+	0.688729i
$134$		−	3.48528i		−	0.301082i
$135$		0			0
$136$	2.12132	−	1.22474i	0.181902	−	0.105021i
$137$	5.19615	−	3.00000i	0.443937	−	0.256307i	−0.261329	−	0.965250i	$-0.584161\pi$
$137$	5.19615	−	3.00000i	0.443937	−	0.256307i	0.705266	+	0.708942i	$0.250827\pi$
$138$		0			0
$139$			11.5300i			0.977963i	0.872294	+	0.488981i	$0.162631\pi$
$139$			11.5300i			0.977963i	−0.872294	+	0.488981i	$0.837369\pi$
$140$	−2.44949	+	6.00000i	−0.207020	+	0.507093i
$141$		0			0
$142$	6.36396	−	11.0227i	0.534052	−	0.925005i
$143$	−8.87039	−	15.3640i	−0.741779	−	1.28480i
$144$		0			0
$145$	−21.7279	−	12.5446i	−1.80441	−	1.04177i
$146$	15.2913			1.26552
$147$		0			0
$148$	3.24264			0.266543
$149$	−6.71807	−	3.87868i	−0.550366	−	0.317754i	0.198904	−	0.980019i	$-0.436262\pi$
$149$	−6.71807	−	3.87868i	−0.550366	−	0.317754i	−0.749270	+	0.662265i	$0.769595\pi$
$150$		0			0
$151$	−8.62132	−	14.9326i	−0.701593	−	1.21519i	−0.967907	−	0.251309i	$-0.919139\pi$
$151$	−8.62132	−	14.9326i	−0.701593	−	1.21519i	0.266314	−	0.963886i	$-0.414194\pi$
$152$	2.44949	−	4.24264i	0.198680	−	0.344124i
$153$		0			0
$154$	4.24264	−	10.3923i	0.341882	−	0.837436i
$155$		−	13.7574i		−	1.10502i
$156$		0			0
$157$	−9.00000	+	5.19615i	−0.718278	+	0.414698i	−0.814119	−	0.580699i	$-0.802779\pi$
$157$	−9.00000	+	5.19615i	−0.718278	+	0.414698i	0.0958404	+	0.995397i	$0.469446\pi$
$158$	8.00436	−	4.62132i	0.636793	−	0.367653i
$159$		0			0
$160$		−	2.44949i		−	0.193649i
$161$	12.5446	−	9.72792i	0.988655	−	0.766668i
$162$		0			0
$163$	−3.74264	+	6.48244i	−0.293146	+	0.507744i	−0.974552	−	0.224162i	$-0.928036\pi$
$163$	−3.74264	+	6.48244i	−0.293146	+	0.507744i	0.681406	+	0.731906i	$0.261369\pi$
$164$	1.22474	+	2.12132i	0.0956365	+	0.165647i
$165$		0			0
$166$	−4.75736	−	2.74666i	−0.369243	−	0.213182i
$167$	−20.1903			−1.56237			−0.781185	−	0.624300i	$-0.785384\pi$
$167$	−20.1903			−1.56237			−0.781185	+	0.624300i	$0.785384\pi$
$168$		0			0
$169$	−4.48528			−0.345022
$170$	−5.19615	−	3.00000i	−0.398527	−	0.230089i
$171$		0			0
$172$	3.50000	+	6.06218i	0.266872	+	0.462237i
$173$	−10.3923	+	18.0000i	−0.790112	+	1.36851i	0.135785	+	0.990738i	$0.456644\pi$
$173$	−10.3923	+	18.0000i	−0.790112	+	1.36851i	−0.925897	+	0.377776i	$0.876689\pi$
$174$		0			0
$175$	2.62132	−	0.358719i	0.198153	−	0.0271166i
$176$			4.24264i			0.319801i
$177$		0			0
$178$	15.3640	−	8.87039i	1.15158	−	0.664864i
$179$	16.2189	−	9.36396i	1.21225	−	0.699895i	0.249004	−	0.968502i	$-0.419897\pi$
$179$	16.2189	−	9.36396i	1.21225	−	0.699895i	0.963250	+	0.268607i	$0.0865633\pi$
$180$		0			0
$181$		−	9.79796i		−	0.728277i	−0.931345	−	0.364138i	$-0.881364\pi$
$181$		−	9.79796i		−	0.728277i	0.931345	−	0.364138i	$-0.118636\pi$
$182$	−6.77962	−	8.74264i	−0.502539	−	0.648048i
$183$		0			0
$184$	−3.00000	+	5.19615i	−0.221163	+	0.383065i
$185$	−3.97141	−	6.87868i	−0.291984	−	0.505731i
$186$		0			0
$187$	9.00000	+	5.19615i	0.658145	+	0.379980i
$188$	−7.94282			−0.579289
$189$		0			0
$190$	−12.0000			−0.870572
$191$	18.3712	+	10.6066i	1.32929	+	0.767467i	0.985190	−	0.171466i	$-0.0548503\pi$
$191$	18.3712	+	10.6066i	1.32929	+	0.767467i	0.344101	+	0.938933i	$0.388184\pi$
$192$		0			0
$193$	−7.74264	−	13.4106i	−0.557327	−	0.965319i	−0.997718	−	0.0675134i	$-0.978493\pi$
$193$	−7.74264	−	13.4106i	−0.557327	−	0.965319i	0.440391	−	0.897806i	$-0.354840\pi$
$194$	3.31552	−	5.74264i	0.238040	−	0.412298i
$195$		0			0
$196$	1.74264	−	6.77962i	0.124474	−	0.484258i
$197$		−	16.9706i		−	1.20910i	−0.796566	−	0.604551i	$-0.793352\pi$
$197$		−	16.9706i		−	1.20910i	0.796566	−	0.604551i	$-0.206648\pi$
$198$		0			0
$199$	3.10660	−	1.79360i	0.220221	−	0.127145i	−0.385832	−	0.922569i	$-0.626085\pi$
$199$	3.10660	−	1.79360i	0.220221	−	0.127145i	0.606053	+	0.795425i	$0.292752\pi$
$200$	−0.866025	+	0.500000i	−0.0612372	+	0.0353553i
$201$		0			0
$202$		−	7.34847i		−	0.517036i
$203$	25.0892	+	10.2426i	1.76092	+	0.718892i
$204$		0			0
$205$	3.00000	−	5.19615i	0.209529	−	0.362915i
$206$	−3.10538	−	5.37868i	−0.216362	−	0.374750i
$207$		0			0
$208$	3.62132	+	2.09077i	0.251093	+	0.144969i
$209$	20.7846			1.43770
$210$		0			0
$211$	−13.4853			−0.928365			−0.464183	−	0.885740i	$-0.653652\pi$
$211$	−13.4853			−0.928365			−0.464183	+	0.885740i	$0.653652\pi$
$212$	−2.15232	−	1.24264i	−0.147822	−	0.0853449i
$213$		0			0
$214$	7.24264	+	12.5446i	0.495097	+	0.857533i
$215$	8.57321	−	14.8492i	0.584688	−	1.01271i
$216$		0			0
$217$	2.01472	+	14.7224i	0.136768	+	0.999424i
$218$			7.72792i			0.523401i
$219$		0			0
$220$	9.00000	−	5.19615i	0.606780	−	0.350325i
$221$	8.87039	−	5.12132i	0.596687	−	0.344497i
$222$		0			0
$223$			10.3923i			0.695920i	0.937509	+	0.347960i	$0.113126\pi$
$223$			10.3923i			0.695920i	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0.358719	+	2.62132i	0.0239680	+	0.175144i
$225$		0			0
$226$	0.878680	−	1.52192i	0.0584489	−	0.101236i
$227$	−2.15232	−	3.72792i	−0.142854	−	0.247431i	0.785716	−	0.618587i	$-0.212295\pi$
$227$	−2.15232	−	3.72792i	−0.142854	−	0.247431i	−0.928570	+	0.371156i	$0.878961\pi$
$228$		0			0
$229$	7.86396	+	4.54026i	0.519665	+	0.300029i	0.736798	−	0.676113i	$-0.236337\pi$
$229$	7.86396	+	4.54026i	0.519665	+	0.300029i	−0.217132	+	0.976142i	$0.569670\pi$
$230$	14.6969			0.969087
$231$		0			0
$232$	−10.2426			−0.672462
$233$	−3.04384	−	1.75736i	−0.199408	−	0.115128i	0.396971	−	0.917831i	$-0.370061\pi$
$233$	−3.04384	−	1.75736i	−0.199408	−	0.115128i	−0.596379	+	0.802703i	$0.703395\pi$
$234$		0			0
$235$	9.72792	+	16.8493i	0.634580	+	1.09912i
$236$	1.22474	−	2.12132i	0.0797241	−	0.138086i
$237$		0			0
$238$	6.00000	+	2.44949i	0.388922	+	0.158777i
$239$			7.75736i			0.501782i	0.968015	+	0.250891i	$0.0807236\pi$
$239$			7.75736i			0.501782i	−0.968015	+	0.250891i	$0.919276\pi$
$240$		0			0
$241$	15.9853	−	9.22911i	1.02970	−	0.594499i	0.112803	−	0.993617i	$-0.464017\pi$
$241$	15.9853	−	9.22911i	1.02970	−	0.594499i	0.916899	+	0.399118i	$0.130684\pi$
$242$	−6.06218	+	3.50000i	−0.389692	+	0.224989i
$243$		0			0
$244$		−	0.717439i		−	0.0459293i
$245$	−16.5160	+	4.60660i	−1.05517	+	0.294305i
$246$		0			0
$247$	10.2426	−	17.7408i	0.651724	−	1.12882i
$248$	−2.80821	−	4.86396i	−0.178321	−	0.308862i
$249$		0			0
$250$	−8.48528	−	4.89898i	−0.536656	−	0.309839i
$251$	−5.49333			−0.346736			−0.173368	−	0.984857i	$-0.555465\pi$
$251$	−5.49333			−0.346736			−0.173368	+	0.984857i	$0.555465\pi$
$252$		0			0
$253$	−25.4558			−1.60040
$254$	−15.3528	−	8.86396i	−0.963322	−	0.556174i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−2.15232	+	3.72792i	−0.134258	+	0.232541i	−0.925314	−	0.379203i	$-0.876198\pi$
$257$	−2.15232	+	3.72792i	−0.134258	+	0.232541i	0.791056	+	0.611744i	$0.209532\pi$
$258$		0			0
$259$	5.25736	+	6.77962i	0.326676	+	0.421265i
$260$		−	10.2426i		−	0.635222i
$261$		0			0
$262$		0			0
$263$	−16.2189	+	9.36396i	−1.00010	+	0.577407i	−0.908277	−	0.418369i	$-0.862602\pi$
$263$	−16.2189	+	9.36396i	−1.00010	+	0.577407i	−0.0918204	+	0.995776i	$0.529269\pi$
$264$		0			0
$265$			6.08767i			0.373963i
$266$	12.8418	−	1.75736i	0.787381	−	0.107751i
$267$		0			0
$268$	−1.74264	+	3.01834i	−0.106449	+	0.184375i
$269$	−4.89898	−	8.48528i	−0.298696	−	0.517357i	0.677142	−	0.735853i	$-0.263218\pi$
$269$	−4.89898	−	8.48528i	−0.298696	−	0.517357i	−0.975838	+	0.218496i	$0.929885\pi$
$270$		0			0
$271$	13.3492	+	7.70719i	0.810909	+	0.468178i	0.847271	−	0.531160i	$-0.178244\pi$
$271$	13.3492	+	7.70719i	0.810909	+	0.468178i	−0.0363626	+	0.999339i	$0.511577\pi$
$272$	−2.44949			−0.148522
$273$		0			0
$274$	−6.00000			−0.362473
$275$	−3.67423	−	2.12132i	−0.221565	−	0.127920i
$276$		0			0
$277$	0.863961	+	1.49642i	0.0519104	+	0.0899114i	0.890813	−	0.454370i	$-0.150136\pi$
$277$	0.863961	+	1.49642i	0.0519104	+	0.0899114i	−0.838903	+	0.544282i	$0.816802\pi$
$278$	5.76500	−	9.98528i	0.345762	−	0.598877i
$279$		0			0
$280$	5.12132	−	3.97141i	0.306057	−	0.237337i
$281$		−	22.2426i		−	1.32688i	−0.748227	−	0.663442i	$-0.769095\pi$
$281$		−	22.2426i		−	1.32688i	0.748227	−	0.663442i	$-0.230905\pi$
$282$		0			0
$283$	−20.2279	+	11.6786i	−1.20243	+	0.694220i	−0.961094	−	0.276222i	$-0.910917\pi$
$283$	−20.2279	+	11.6786i	−1.20243	+	0.694220i	−0.241331	+	0.970443i	$0.577584\pi$
$284$	−11.0227	+	6.36396i	−0.654077	+	0.377632i
$285$		0			0
$286$			17.7408i			1.04903i
$287$	−2.44949	+	6.00000i	−0.144589	+	0.354169i
$288$		0			0
$289$	5.50000	−	9.52628i	0.323529	−	0.560369i
$290$	12.5446	+	21.7279i	0.736646	+	1.27591i
$291$		0			0
$292$	−13.2426	−	7.64564i	−0.774967	−	0.447427i
$293$	−7.94282			−0.464024			−0.232012	−	0.972713i	$-0.574531\pi$
$293$	−7.94282			−0.464024			−0.232012	+	0.972713i	$0.574531\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$	−2.80821	−	1.62132i	−0.163224	−	0.0942373i
$297$		0			0
$298$	3.87868	+	6.71807i	0.224686	+	0.389167i
$299$	−12.5446	+	21.7279i	−0.725474	+	1.25656i
$300$		0			0
$301$	−7.00000	+	17.1464i	−0.403473	+	0.988304i
$302$			17.2426i			0.992202i
$303$		0			0
$304$	−4.24264	+	2.44949i	−0.243332	+	0.140488i
$305$	−1.52192	+	0.878680i	−0.0871448	+	0.0503131i
$306$		0			0
$307$			2.57258i			0.146825i	0.997302	+	0.0734125i	$0.0233890\pi$
$307$			2.57258i			0.146825i	−0.997302	+	0.0734125i	$0.976611\pi$
$308$	−8.87039	+	6.87868i	−0.505437	+	0.391949i
$309$		0			0
$310$	−6.87868	+	11.9142i	−0.390683	+	0.676682i
$311$	8.57321	+	14.8492i	0.486142	+	0.842023i	0.999873	−	0.0159282i	$-0.00507031\pi$
$311$	8.57321	+	14.8492i	0.486142	+	0.842023i	−0.513731	+	0.857951i	$0.671737\pi$
$312$		0			0
$313$	−0.514719	−	0.297173i	−0.0290936	−	0.0167972i	0.485383	−	0.874302i	$-0.338680\pi$
$313$	−0.514719	−	0.297173i	−0.0290936	−	0.0167972i	−0.514476	+	0.857505i	$0.672014\pi$
$314$	10.3923			0.586472
$315$		0			0
$316$	−9.24264			−0.519939
$317$	11.9142	+	6.87868i	0.669169	+	0.386345i	0.795762	−	0.605610i	$-0.207071\pi$
$317$	11.9142	+	6.87868i	0.669169	+	0.386345i	−0.126592	+	0.991955i	$0.540404\pi$
$318$		0			0
$319$	−21.7279	−	37.6339i	−1.21653	−	2.10709i
$320$	−1.22474	+	2.12132i	−0.0684653	+	0.118585i
$321$		0			0
$322$	−15.7279	+	2.15232i	−0.876483	+	0.119944i
$323$			12.0000i			0.667698i
$324$		0			0
$325$	−3.62132	+	2.09077i	−0.200875	+	0.115975i
$326$	6.48244	−	3.74264i	0.359029	−	0.207286i
$327$		0			0
$328$		−	2.44949i		−	0.135250i
$329$	−12.8778	−	16.6066i	−0.709979	−	0.915552i
$330$		0			0
$331$	−5.00000	+	8.66025i	−0.274825	+	0.476011i	−0.970091	−	0.242742i	$-0.921953\pi$
$331$	−5.00000	+	8.66025i	−0.274825	+	0.476011i	0.695266	+	0.718752i	$0.255287\pi$
$332$	2.74666	+	4.75736i	0.150743	+	0.261094i
$333$		0			0
$334$	17.4853	+	10.0951i	0.956752	+	0.552381i
$335$	8.53716			0.466435
$336$		0			0
$337$	29.4558			1.60456			0.802281	−	0.596947i	$-0.203620\pi$
$337$	29.4558			1.60456			0.802281	+	0.596947i	$0.203620\pi$
$338$	3.88437	+	2.24264i	0.211282	+	0.121984i
$339$		0			0
$340$	3.00000	+	5.19615i	0.162698	+	0.281801i
$341$	11.9142	−	20.6360i	0.645191	−	1.11750i
$342$		0			0
$343$	17.0000	−	7.34847i	0.917914	−	0.396780i
$344$		−	7.00000i		−	0.377415i
$345$		0			0
$346$	18.0000	−	10.3923i	0.967686	−	0.558694i
$347$	−25.0892	+	14.4853i	−1.34686	+	0.777611i	−0.987804	−	0.155705i	$-0.950235\pi$
$347$	−25.0892	+	14.4853i	−1.34686	+	0.777611i	−0.359058	+	0.933315i	$0.616902\pi$
$348$		0			0
$349$		−	24.3718i		−	1.30459i	−0.757964	−	0.652296i	$-0.773806\pi$
$349$		−	24.3718i		−	1.30459i	0.757964	−	0.652296i	$-0.226194\pi$
$350$	−2.44949	−	1.00000i	−0.130931	−	0.0534522i
$351$		0			0
$352$	2.12132	−	3.67423i	0.113067	−	0.195837i
$353$	15.2913	+	26.4853i	0.813873	+	1.40967i	0.910134	+	0.414313i	$0.135978\pi$
$353$	15.2913	+	26.4853i	0.813873	+	1.40967i	−0.0962614	+	0.995356i	$0.530688\pi$
$354$		0			0
$355$	27.0000	+	15.5885i	1.43301	+	0.827349i
$356$	−17.7408			−0.940259
$357$		0			0
$358$	−18.7279			−0.989801
$359$	−2.15232	−	1.24264i	−0.113595	−	0.0655841i	0.442126	−	0.896953i	$-0.354224\pi$
$359$	−2.15232	−	1.24264i	−0.113595	−	0.0655841i	−0.555721	+	0.831369i	$0.687558\pi$
$360$		0			0
$361$	2.50000	+	4.33013i	0.131579	+	0.227901i
$362$	−4.89898	+	8.48528i	−0.257485	+	0.445976i
$363$		0			0
$364$	1.50000	+	10.9612i	0.0786214	+	0.574521i
$365$			37.4558i			1.96053i
$366$		0			0
$367$	7.97056	−	4.60181i	0.416060	−	0.240212i	−0.277330	−	0.960775i	$-0.589450\pi$
$367$	7.97056	−	4.60181i	0.416060	−	0.240212i	0.693390	+	0.720562i	$0.256116\pi$
$368$	5.19615	−	3.00000i	0.270868	−	0.156386i
$369$		0			0
$370$			7.94282i			0.412927i
$371$	−0.891519	−	6.51472i	−0.0462854	−	0.338227i
$372$		0			0
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	−0.427051	−	0.904227i	$-0.640448\pi$
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	0.996610	−	0.0822766i	$-0.0262191\pi$
$374$	−5.19615	−	9.00000i	−0.268687	−	0.465379i
$375$		0			0
$376$	6.87868	+	3.97141i	0.354741	+	0.204810i
$377$	−42.8300			−2.20586
$378$		0			0
$379$	9.48528			0.487226			0.243613	−	0.969872i	$-0.421667\pi$
$379$	9.48528			0.487226			0.243613	+	0.969872i	$0.421667\pi$
$380$	10.3923	+	6.00000i	0.533114	+	0.307794i
$381$		0			0
$382$	−10.6066	−	18.3712i	−0.542681	−	0.939951i
$383$	7.64564	−	13.2426i	0.390674	−	0.676667i	−0.601865	−	0.798598i	$-0.705575\pi$
$383$	7.64564	−	13.2426i	0.390674	−	0.676667i	0.992539	+	0.121931i	$0.0389086\pi$
$384$		0			0
$385$	25.4558	+	10.3923i	1.29735	+	0.529641i
$386$			15.4853i			0.788180i
$387$		0			0
$388$	−5.74264	+	3.31552i	−0.291538	+	0.168320i
$389$	−28.1331	+	16.2426i	−1.42640	+	0.823535i	−0.996835	−	0.0794995i	$-0.974668\pi$
$389$	−28.1331	+	16.2426i	−1.42640	+	0.823535i	−0.429569	+	0.903034i	$0.641334\pi$
$390$		0			0
$391$		−	14.6969i		−	0.743256i
$392$	−4.89898	+	5.00000i	−0.247436	+	0.252538i
$393$		0			0
$394$	−8.48528	+	14.6969i	−0.427482	+	0.740421i
$395$	11.3199	+	19.6066i	0.569565	+	0.986515i
$396$		0			0
$397$	−25.8640	−	14.9326i	−1.29807	−	0.749444i	−0.318004	−	0.948090i	$-0.603012\pi$
$397$	−25.8640	−	14.9326i	−1.29807	−	0.749444i	−0.980071	+	0.198646i	$0.936346\pi$
$398$	−3.58719			−0.179810
$399$		0			0
$400$	1.00000			0.0500000
$401$	24.4588	+	14.1213i	1.22142	+	0.705185i	0.965220	−	0.261440i	$-0.0841972\pi$
$401$	24.4588	+	14.1213i	1.22142	+	0.705185i	0.256197	+	0.966625i	$0.417531\pi$
$402$		0			0
$403$	−11.7426	−	20.3389i	−0.584943	−	1.01315i
$404$	−3.67423	+	6.36396i	−0.182800	+	0.316619i
$405$		0			0
$406$	−16.6066	−	21.4150i	−0.824172	−	1.06281i
$407$		−	13.7574i		−	0.681927i
$408$		0			0
$409$	−11.7426	+	6.77962i	−0.580636	+	0.335230i	−0.761386	−	0.648299i	$-0.775481\pi$
$409$	−11.7426	+	6.77962i	−0.580636	+	0.335230i	0.180750	+	0.983529i	$0.442148\pi$
$410$	−5.19615	+	3.00000i	−0.256620	+	0.148159i
$411$		0			0
$412$			6.21076i			0.305982i
$413$	6.42090	−	0.878680i	0.315952	−	0.0432370i
$414$		0			0
$415$	6.72792	−	11.6531i	0.330261	−	0.572028i
$416$	−2.09077	−	3.62132i	−0.102508	−	0.177550i
$417$		0			0
$418$	−18.0000	−	10.3923i	−0.880409	−	0.508304i
$419$	−12.8418			−0.627363			−0.313681	−	0.949528i	$-0.601562\pi$
$419$	−12.8418			−0.627363			−0.313681	+	0.949528i	$0.601562\pi$
$420$		0			0
$421$	27.4558			1.33812			0.669058	−	0.743210i	$-0.266698\pi$
$421$	27.4558			1.33812			0.669058	+	0.743210i	$0.266698\pi$
$422$	11.6786	+	6.74264i	0.568505	+	0.328227i
$423$		0			0
$424$	1.24264	+	2.15232i	0.0603480	+	0.104526i
$425$	1.22474	−	2.12132i	0.0594089	−	0.102899i
$426$		0			0
$427$	1.50000	−	1.16320i	0.0725901	−	0.0562911i
$428$		−	14.4853i		−	0.700173i
$429$		0			0
$430$	−14.8492	+	8.57321i	−0.716094	+	0.413437i
$431$	−8.87039	+	5.12132i	−0.427272	+	0.246685i	−0.698184	−	0.715919i	$-0.746008\pi$
$431$	−8.87039	+	5.12132i	−0.427272	+	0.246685i	0.270912	+	0.962604i	$0.412675\pi$
$432$		0			0
$433$			26.8213i			1.28895i	0.764626	+	0.644475i	$0.222924\pi$
$433$			26.8213i			1.28895i	−0.764626	+	0.644475i	$0.777076\pi$
$434$	5.61642	−	13.7574i	0.269597	−	0.660374i
$435$		0			0
$436$	3.86396	−	6.69258i	0.185050	−	0.320516i
$437$	−14.6969	−	25.4558i	−0.703050	−	1.21772i
$438$		0			0
$439$	−21.7279	−	12.5446i	−1.03702	−	0.598722i	−0.118030	−	0.993010i	$-0.537658\pi$
$439$	−21.7279	−	12.5446i	−1.03702	−	0.598722i	−0.918987	+	0.394288i	$0.870991\pi$
$440$	−10.3923			−0.495434
$441$		0			0
$442$	−10.2426			−0.487193
$443$	−22.0454	−	12.7279i	−1.04741	−	0.604722i	−0.125486	−	0.992095i	$-0.540049\pi$
$443$	−22.0454	−	12.7279i	−1.04741	−	0.604722i	−0.921923	+	0.387374i	$0.873382\pi$
$444$		0			0
$445$	21.7279	+	37.6339i	1.03000	+	1.78402i
$446$	5.19615	−	9.00000i	0.246045	−	0.426162i
$447$		0			0
$448$	1.00000	−	2.44949i	0.0472456	−	0.115728i
$449$			30.7279i			1.45014i	0.688675	+	0.725070i	$0.258193\pi$
$449$			30.7279i			1.45014i	−0.688675	+	0.725070i	$0.741807\pi$
$450$		0			0
$451$	9.00000	−	5.19615i	0.423793	−	0.244677i
$452$	−1.52192	+	0.878680i	−0.0715850	+	0.0413296i
$453$		0			0
$454$			4.30463i			0.202026i
$455$	21.4150	−	16.6066i	1.00395	−	0.778529i
$456$		0			0
$457$	11.5000	−	19.9186i	0.537947	−	0.931752i	−0.461067	−	0.887365i	$-0.652533\pi$
$457$	11.5000	−	19.9186i	0.537947	−	0.931752i	0.999014	−	0.0443868i	$-0.0141334\pi$
$458$	−4.54026	−	7.86396i	−0.212152	−	0.367459i
$459$		0			0
$460$	−12.7279	−	7.34847i	−0.593442	−	0.342624i
$461$	42.2357			1.96711			0.983556	−	0.180605i	$-0.0578056\pi$
$461$	42.2357			1.96711			0.983556	+	0.180605i	$0.0578056\pi$
$462$		0			0
$463$	−22.0000			−1.02243			−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000			−1.02243			−0.511213	+	0.859454i	$0.670804\pi$
$464$	8.87039	+	5.12132i	0.411797	+	0.237751i
$465$		0			0
$466$	1.75736	+	3.04384i	0.0814081	+	0.141003i
$467$	1.52192	−	2.63604i	0.0704260	−	0.121981i	−0.828662	−	0.559749i	$-0.810897\pi$
$467$	1.52192	−	2.63604i	0.0704260	−	0.121981i	0.899088	+	0.437768i	$0.144231\pi$
$468$		0			0
$469$	−9.13604	+	1.25024i	−0.421863	+	0.0577306i
$470$		−	19.4558i		−	0.897431i
$471$		0			0
$472$	−2.12132	+	1.22474i	−0.0976417	+	0.0563735i
$473$	25.7196	−	14.8492i	1.18259	−	0.682769i
$474$		0			0
$475$		−	4.89898i		−	0.224781i
$476$	−3.97141	−	5.12132i	−0.182029	−	0.234735i
$477$		0			0
$478$	3.87868	−	6.71807i	0.177407	−	0.307277i
$479$	−1.22474	−	2.12132i	−0.0559600	−	0.0969256i	0.836688	−	0.547679i	$-0.184489\pi$
$479$	−1.22474	−	2.12132i	−0.0559600	−	0.0969256i	−0.892648	+	0.450754i	$0.851155\pi$
$480$		0			0
$481$	−11.7426	−	6.77962i	−0.535418	−	0.309124i
$482$	−18.4582			−0.840749
$483$		0			0
$484$	7.00000			0.318182
$485$	14.0665	+	8.12132i	0.638729	+	0.368770i
$486$		0			0
$487$	−11.0000	−	19.0526i	−0.498458	−	0.863354i	0.501541	−	0.865134i	$-0.332767\pi$
$487$	−11.0000	−	19.0526i	−0.498458	−	0.863354i	−0.999998	+	0.00178012i	$0.999433\pi$
$488$	−0.358719	+	0.621320i	−0.0162385	+	0.0281259i
$489$		0			0
$490$	16.6066	+	4.26858i	0.750210	+	0.192835i
$491$			1.02944i			0.0464579i	0.999730	+	0.0232289i	$0.00739466\pi$
$491$			1.02944i			0.0464579i	−0.999730	+	0.0232289i	$0.992605\pi$
$492$		0			0
$493$	21.7279	−	12.5446i	0.978576	−	0.564981i
$494$	−17.7408	+	10.2426i	−0.798195	+	0.460838i
$495$		0			0
$496$			5.61642i			0.252185i
$497$	−31.1769	−	12.7279i	−1.39848	−	0.570925i
$498$		0			0
$499$	−13.2279	+	22.9114i	−0.592163	+	1.02566i	0.401777	+	0.915737i	$0.368393\pi$
$499$	−13.2279	+	22.9114i	−0.592163	+	1.02566i	−0.993941	+	0.109919i	$0.964941\pi$
$500$	4.89898	+	8.48528i	0.219089	+	0.379473i
$501$		0			0
$502$	4.75736	+	2.74666i	0.212331	+	0.122590i
$503$	20.1903			0.900239			0.450120	−	0.892968i	$-0.351381\pi$
$503$	20.1903			0.900239			0.450120	+	0.892968i	$0.351381\pi$
$504$		0			0
$505$	18.0000			0.800989
$506$	22.0454	+	12.7279i	0.980038	+	0.565825i
$507$		0			0
$508$	8.86396	+	15.3528i	0.393275	+	0.681172i
$509$	14.9941	−	25.9706i	0.664602	−	1.15112i	−0.314791	−	0.949161i	$-0.601934\pi$
$509$	14.9941	−	25.9706i	0.664602	−	1.15112i	0.979393	−	0.201964i	$-0.0647323\pi$
$510$		0			0
$511$	−5.48528	−	40.0834i	−0.242655	−	1.77318i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	3.72792	−	2.15232i	0.164432	−	0.0949346i
$515$	13.1750	−	7.60660i	0.580561	−	0.335187i
$516$		0			0
$517$			33.6985i			1.48206i
$518$	−1.16320	−	8.50000i	−0.0511080	−	0.373469i
$519$		0			0
$520$	−5.12132	+	8.87039i	−0.224585	+	0.388992i
$521$	4.60181	+	7.97056i	0.201609	+	0.349197i	0.949047	−	0.315135i	$-0.102050\pi$
$521$	4.60181	+	7.97056i	0.201609	+	0.349197i	−0.747438	+	0.664331i	$0.768716\pi$
$522$		0			0
$523$	15.2574	+	8.80884i	0.667158	+	0.385184i	0.794999	−	0.606611i	$-0.207471\pi$
$523$	15.2574	+	8.80884i	0.667158	+	0.385184i	−0.127841	+	0.991795i	$0.540805\pi$
$524$		0			0
$525$		0			0
$526$	18.7279			0.816576
$527$	11.9142	+	6.87868i	0.518992	+	0.299640i
$528$		0			0
$529$	6.50000	+	11.2583i	0.282609	+	0.489493i
$530$	3.04384	−	5.27208i	0.132216	−	0.229004i
$531$		0			0
$532$	−12.0000	−	4.89898i	−0.520266	−	0.212398i
$533$		−	10.2426i		−	0.443658i
$534$		0			0
$535$	−30.7279	+	17.7408i	−1.32848	+	0.767001i
$536$	3.01834	−	1.74264i	0.130373	−	0.0752706i
$537$		0			0
$538$			9.79796i			0.422420i
$539$	−28.7635	−	7.39340i	−1.23893	−	0.318456i
$540$		0			0
$541$	−22.7279	+	39.3659i	−0.977150	+	1.69247i	−0.304499	+	0.952513i	$0.598489\pi$
$541$	−22.7279	+	39.3659i	−0.977150	+	1.69247i	−0.672651	+	0.739960i	$0.734844\pi$
$542$	−7.70719	−	13.3492i	−0.331052	−	0.573399i
$543$		0			0
$544$	2.12132	+	1.22474i	0.0909509	+	0.0525105i
$545$	−18.9295			−0.810849
$546$		0			0
$547$	6.02944			0.257800			0.128900	−	0.991658i	$-0.458855\pi$
$547$	6.02944			0.257800			0.128900	+	0.991658i	$0.458855\pi$
$548$	5.19615	+	3.00000i	0.221969	+	0.128154i
$549$		0			0
$550$	2.12132	+	3.67423i	0.0904534	+	0.156670i
$551$	25.0892	−	43.4558i	1.06884	−	1.85128i
$552$		0			0
$553$	−14.9853	−	19.3242i	−0.637239	−	0.821750i
$554$		−	1.72792i		−	0.0734124i
$555$		0			0
$556$	−9.98528	+	5.76500i	−0.423470	+	0.244491i
$557$	−18.3712	+	10.6066i	−0.778412	+	0.449416i	−0.835867	−	0.548932i	$-0.815035\pi$
$557$	−18.3712	+	10.6066i	−0.778412	+	0.449416i	0.0574555	+	0.998348i	$0.481701\pi$
$558$		0			0
$559$		−	29.2708i		−	1.23802i
$560$	−6.42090	+	0.878680i	−0.271332	+	0.0371310i
$561$		0			0
$562$	−11.1213	+	19.2627i	−0.469125	+	0.812548i
$563$	−8.23999	−	14.2721i	−0.347274	−	0.601496i	0.638490	−	0.769630i	$-0.279559\pi$
$563$	−8.23999	−	14.2721i	−0.347274	−	0.601496i	−0.985764	+	0.168134i	$0.946226\pi$
$564$		0			0
$565$	3.72792	+	2.15232i	0.156835	+	0.0905486i
$566$	23.3572			0.981776
$567$		0			0
$568$	12.7279			0.534052
$569$	1.52192	+	0.878680i	0.0638021	+	0.0368362i	0.531562	−	0.847020i	$-0.321605\pi$
$569$	1.52192	+	0.878680i	0.0638021	+	0.0368362i	−0.467760	+	0.883856i	$0.654939\pi$
$570$		0			0
$571$	11.0000	+	19.0526i	0.460336	+	0.797325i	0.998978	−	0.0452101i	$-0.0143957\pi$
$571$	11.0000	+	19.0526i	0.460336	+	0.797325i	−0.538642	+	0.842535i	$0.681062\pi$
$572$	8.87039	−	15.3640i	0.370890	−	0.642399i
$573$		0			0
$574$	5.12132	−	3.97141i	0.213760	−	0.165763i
$575$			6.00000i			0.250217i
$576$		0			0
$577$	15.2574	−	8.80884i	0.635172	−	0.366717i	−0.147580	−	0.989050i	$-0.547148\pi$
$577$	15.2574	−	8.80884i	0.635172	−	0.366717i	0.782752	+	0.622333i	$0.213815\pi$
$578$	−9.52628	+	5.50000i	−0.396241	+	0.228770i
$579$		0			0
$580$		−	25.0892i		−	1.04177i
$581$	−5.49333	+	13.4558i	−0.227902	+	0.558242i
$582$		0			0
$583$	−5.27208	+	9.13151i	−0.218347	+	0.378188i
$584$	7.64564	+	13.2426i	0.316379	+	0.547984i
$585$		0			0
$586$	6.87868	+	3.97141i	0.284156	+	0.164057i
$587$	11.6531			0.480975			0.240488	−	0.970652i	$-0.422693\pi$
$587$	11.6531			0.480975			0.240488	+	0.970652i	$0.422693\pi$
$588$		0			0
$589$	27.5147			1.13372
$590$	5.19615	+	3.00000i	0.213922	+	0.123508i
$591$		0			0
$592$	1.62132	+	2.80821i	0.0666359	+	0.115417i
$593$	11.3199	−	19.6066i	0.464852	−	0.805147i	−0.534343	−	0.845268i	$-0.679441\pi$
$593$	11.3199	−	19.6066i	0.464852	−	0.805147i	0.999195	+	0.0401210i	$0.0127743\pi$
$594$		0			0
$595$	−6.00000	+	14.6969i	−0.245976	+	0.602516i
$596$		−	7.75736i		−	0.317754i
$597$		0			0
$598$	21.7279	−	12.5446i	0.888521	−	0.512988i
$599$	−17.7408	+	10.2426i	−0.724868	+	0.418503i	−0.816542	−	0.577286i	$-0.804112\pi$
$599$	−17.7408	+	10.2426i	−0.724868	+	0.418503i	0.0916735	+	0.995789i	$0.470778\pi$
$600$		0			0
$601$			17.2695i			0.704438i	0.935918	+	0.352219i	$0.114573\pi$
$601$			17.2695i			0.704438i	−0.935918	+	0.352219i	$0.885427\pi$
$602$	14.6354	−	11.3492i	0.596494	−	0.462561i
$603$		0			0
$604$	8.62132	−	14.9326i	0.350797	−	0.607597i
$605$	−8.57321	−	14.8492i	−0.348551	−	0.603708i
$606$		0			0
$607$	22.7574	+	13.1390i	0.923693	+	0.533294i	0.884811	−	0.465950i	$-0.154287\pi$
$607$	22.7574	+	13.1390i	0.923693	+	0.533294i	0.0388815	+	0.999244i	$0.487621\pi$
$608$	4.89898			0.198680
$609$		0			0
$610$	1.75736			0.0711534
$611$	28.7635	+	16.6066i	1.16365	+	0.671831i
$612$		0			0
$613$	−7.10660	−	12.3090i	−0.287033	−	0.497156i	0.686067	−	0.727538i	$-0.259336\pi$
$613$	−7.10660	−	12.3090i	−0.287033	−	0.497156i	−0.973100	+	0.230383i	$0.926002\pi$
$614$	1.28629	−	2.22792i	0.0519105	−	0.0899116i
$615$		0			0
$616$	11.1213	−	1.52192i	0.448091	−	0.0613198i
$617$		−	4.24264i		−	0.170802i	−0.996347	−	0.0854011i	$-0.972783\pi$
$617$		−	4.24264i		−	0.170802i	0.996347	−	0.0854011i	$-0.0272172\pi$
$618$		0			0
$619$	−27.9853	+	16.1573i	−1.12482	+	0.649417i	−0.942628	−	0.333845i	$-0.891654\pi$
$619$	−27.9853	+	16.1573i	−1.12482	+	0.649417i	−0.182196	+	0.983262i	$0.558320\pi$
$620$	11.9142	−	6.87868i	0.478487	−	0.276254i
$621$		0			0
$622$		−	17.1464i		−	0.687509i
$623$	−28.7635	−	37.0919i	−1.15238	−	1.48605i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$	0.297173	+	0.514719i	0.0118774	+	0.0205723i
$627$		0			0
$628$	−9.00000	−	5.19615i	−0.359139	−	0.207349i
$629$	7.94282			0.316701
$630$		0			0
$631$	−23.2426			−0.925275			−0.462637	−	0.886548i	$-0.653097\pi$
$631$	−23.2426			−0.925275			−0.462637	+	0.886548i	$0.653097\pi$
$632$	8.00436	+	4.62132i	0.318396	+	0.183826i
$633$		0			0
$634$	−6.87868	−	11.9142i	−0.273187	−	0.473174i
$635$	21.7122	−	37.6066i	0.861622	−	1.49237i
$636$		0			0
$637$	−20.4853	+	20.9077i	−0.811656	+	0.828393i
$638$			43.4558i			1.72043i
$639$		0			0
$640$	2.12132	−	1.22474i	0.0838525	−	0.0484123i
$641$	7.97887	−	4.60660i	0.315146	−	0.181950i	−0.334081	−	0.942544i	$-0.608426\pi$
$641$	7.97887	−	4.60660i	0.315146	−	0.181950i	0.649227	+	0.760595i	$0.275092\pi$
$642$		0			0
$643$			1.73205i			0.0683054i	0.999417	+	0.0341527i	$0.0108733\pi$
$643$			1.73205i			0.0683054i	−0.999417	+	0.0341527i	$0.989127\pi$
$644$	14.6969	+	6.00000i	0.579141	+	0.236433i
$645$		0			0
$646$	6.00000	−	10.3923i	0.236067	−	0.408880i
$647$	10.3923	+	18.0000i	0.408564	+	0.707653i	0.994729	−	0.102538i	$-0.0326965\pi$
$647$	10.3923	+	18.0000i	0.408564	+	0.707653i	−0.586165	+	0.810191i	$0.699363\pi$
$648$		0			0
$649$	−9.00000	−	5.19615i	−0.353281	−	0.203967i
$650$	4.18154			0.164014
$651$		0			0
$652$	−7.48528			−0.293146
$653$	−12.5446	−	7.24264i	−0.490909	−	0.283426i	0.234043	−	0.972226i	$-0.424805\pi$
$653$	−12.5446	−	7.24264i	−0.490909	−	0.283426i	−0.724951	+	0.688800i	$0.758138\pi$
$654$		0			0
$655$		0			0
$656$	−1.22474	+	2.12132i	−0.0478183	+	0.0828236i
$657$		0			0
$658$	2.84924	+	20.8207i	0.111075	+	0.811674i
$659$			13.7574i			0.535911i	0.963431	+	0.267955i	$0.0863480\pi$
$659$			13.7574i			0.535911i	−0.963431	+	0.267955i	$0.913652\pi$
$660$		0			0
$661$	4.24264	−	2.44949i	0.165020	−	0.0952741i	−0.415216	−	0.909723i	$-0.636294\pi$
$661$	4.24264	−	2.44949i	0.165020	−	0.0952741i	0.580235	+	0.814449i	$0.302961\pi$
$662$	8.66025	−	5.00000i	0.336590	−	0.194331i
$663$		0			0
$664$		−	5.49333i		−	0.213182i
$665$	4.30463	+	31.4558i	0.166927	+	1.21981i
$666$		0			0
$667$	−30.7279	+	53.2223i	−1.18979	+	2.06078i
$668$	−10.0951	−	17.4853i	−0.390592	−	0.676526i
$669$		0			0
$670$	−7.39340	−	4.26858i	−0.285632	−	0.164910i
$671$	−3.04384			−0.117506
$672$		0			0
$673$	5.45584			0.210307			0.105154	−	0.994456i	$-0.466467\pi$
$673$	5.45584			0.210307			0.105154	+	0.994456i	$0.466467\pi$
$674$	−25.5095	−	14.7279i	−0.982590	−	0.567298i
$675$		0			0
$676$	−2.24264	−	3.88437i	−0.0862554	−	0.149399i
$677$	7.34847	−	12.7279i	0.282425	−	0.489174i	−0.689557	−	0.724232i	$-0.742195\pi$
$677$	7.34847	−	12.7279i	0.282425	−	0.489174i	0.971981	+	0.235058i	$0.0755280\pi$
$678$		0			0
$679$	−16.2426	−	6.63103i	−0.623335	−	0.254476i
$680$		−	6.00000i		−	0.230089i
$681$		0			0
$682$	−20.6360	+	11.9142i	−0.790195	+	0.456219i
$683$	−1.52192	+	0.878680i	−0.0582346	+	0.0336217i	−0.528835	−	0.848725i	$-0.677371\pi$
$683$	−1.52192	+	0.878680i	−0.0582346	+	0.0336217i	0.470600	+	0.882347i	$0.344037\pi$
$684$		0			0
$685$		−	14.6969i		−	0.561541i
$686$	−18.3967	−	2.13604i	−0.702388	−	0.0815543i
$687$		0			0
$688$	−3.50000	+	6.06218i	−0.133436	+	0.231118i
$689$	5.19615	+	9.00000i	0.197958	+	0.342873i
$690$		0			0
$691$	−23.2279	−	13.4106i	−0.883632	−	0.510165i	−0.0117776	−	0.999931i	$-0.503749\pi$
$691$	−23.2279	−	13.4106i	−0.883632	−	0.510165i	−0.871854	+	0.489766i	$0.837082\pi$
$692$	−20.7846			−0.790112
$693$		0			0
$694$	28.9706			1.09971
$695$	24.4588	+	14.1213i	0.927777	+	0.535652i
$696$		0			0
$697$	3.00000	+	5.19615i	0.113633	+	0.196818i
$698$	−12.1859	+	21.1066i	−0.461243	+	0.798897i
$699$		0			0
$700$	1.62132	+	2.09077i	0.0612801	+	0.0790237i
$701$			3.51472i			0.132749i	0.997795	+	0.0663745i	$0.0211432\pi$
$701$			3.51472i			0.132749i	−0.997795	+	0.0663745i	$0.978857\pi$
$702$		0			0
$703$	13.7574	−	7.94282i	0.518869	−	0.299569i
$704$	−3.67423	+	2.12132i	−0.138478	+	0.0799503i
$705$		0			0
$706$		−	30.5826i		−	1.15099i
$707$	−19.2627	+	2.63604i	−0.724448	+	0.0991384i
$708$		0			0
$709$	13.1066	−	22.7013i	0.492229	−	0.852565i	−0.507731	−	0.861516i	$-0.669516\pi$
$709$	13.1066	−	22.7013i	0.492229	−	0.852565i	0.999960	+	0.00895033i	$0.00284902\pi$
$710$	−15.5885	−	27.0000i	−0.585024	−	1.01329i
$711$		0			0
$712$	15.3640	+	8.87039i	0.575789	+	0.332432i
$713$	−33.6985			−1.26202
$714$		0			0
$715$	−43.4558			−1.62516
$716$	16.2189	+	9.36396i	0.606127	+	0.349948i
$717$		0			0
$718$	1.24264	+	2.15232i	0.0463749	+	0.0803237i
$719$	−5.52938	+	9.57716i	−0.206211	+	0.357168i	−0.950518	−	0.310670i	$-0.899447\pi$
$719$	−5.52938	+	9.57716i	−0.206211	+	0.357168i	0.744307	+	0.667838i	$0.232780\pi$
$720$		0			0
$721$	−12.9853	+	10.0696i	−0.483597	+	0.375013i
$722$		−	5.00000i		−	0.186081i
$723$		0			0
$724$	8.48528	−	4.89898i	0.315353	−	0.182069i
$725$	−8.87039	+	5.12132i	−0.329438	+	0.190201i
$726$		0			0
$727$		−	45.4026i		−	1.68389i	−0.539564	−	0.841945i	$-0.681411\pi$
$727$		−	45.4026i		−	1.68389i	0.539564	−	0.841945i	$-0.318589\pi$
$728$	4.18154	−	10.2426i	0.154978	−	0.379618i
$729$		0			0
$730$	18.7279	−	32.4377i	0.693151	−	1.20057i
$731$	8.57321	+	14.8492i	0.317092	+	0.549219i
$732$		0			0
$733$	13.8640	+	8.00436i	0.512077	+	0.295648i	0.733687	−	0.679488i	$-0.237798\pi$
$733$	13.8640	+	8.00436i	0.512077	+	0.295648i	−0.221610	+	0.975135i	$0.571131\pi$
$734$	−9.20361			−0.339712
$735$		0			0
$736$	−6.00000			−0.221163
$737$	12.8057	+	7.39340i	0.471706	+	0.272339i
$738$		0			0
$739$	−21.2279	−	36.7678i	−0.780882	−	1.35253i	−0.931429	−	0.363924i	$-0.881437\pi$
$739$	−21.2279	−	36.7678i	−0.780882	−	1.35253i	0.150547	−	0.988603i	$-0.451897\pi$
$740$	3.97141	−	6.87868i	0.145992	−	0.252865i
$741$		0			0
$742$	−2.48528	+	6.08767i	−0.0912375	+	0.223485i
$743$		−	6.72792i		−	0.246824i	−0.992356	−	0.123412i	$-0.960616\pi$
$743$		−	6.72792i		−	0.246824i	0.992356	−	0.123412i	$-0.0393836\pi$
$744$		0			0
$745$	−16.4558	+	9.50079i	−0.602895	+	0.348082i
$746$	−19.0526	+	11.0000i	−0.697564	+	0.402739i
$747$		0			0
$748$			10.3923i			0.379980i
$749$	30.2854	−	23.4853i	1.10660	−	0.858134i
$750$		0			0
$751$	−1.27208	+	2.20330i	−0.0464188	+	0.0803997i	−0.888301	−	0.459261i	$-0.848114\pi$
$751$	−1.27208	+	2.20330i	−0.0464188	+	0.0803997i	0.841882	+	0.539661i	$0.181448\pi$
$752$	−3.97141	−	6.87868i	−0.144822	−	0.250840i
$753$		0			0
$754$	37.0919	+	21.4150i	1.35081	+	0.779889i
$755$	−42.2357			−1.53711
$756$		0			0
$757$	41.2426			1.49899			0.749495	−	0.662010i	$-0.230297\pi$
$757$	41.2426			1.49899			0.749495	+	0.662010i	$0.230297\pi$
$758$	−8.21449	−	4.74264i	−0.298364	−	0.172260i
$759$		0			0
$760$	−6.00000	−	10.3923i	−0.217643	−	0.376969i
$761$	−12.2474	+	21.2132i	−0.443970	+	0.768978i	−0.997980	−	0.0635319i	$-0.979764\pi$
$761$	−12.2474	+	21.2132i	−0.443970	+	0.768978i	0.554010	+	0.832510i	$0.313097\pi$
$762$		0			0
$763$	20.2574	−	2.77216i	0.733366	−	0.100359i
$764$			21.2132i			0.767467i
$765$		0			0
$766$	−13.2426	+	7.64564i	−0.478476	+	0.276248i
$767$	−8.87039	+	5.12132i	−0.320291	+	0.184920i
$768$		0			0
$769$			1.18869i			0.0428653i	0.999770	+	0.0214327i	$0.00682275\pi$
$769$			1.18869i			0.0428653i	−0.999770	+	0.0214327i	$0.993177\pi$
$770$	−16.8493	−	21.7279i	−0.607205	−	0.783020i
$771$		0			0
$772$	7.74264	−	13.4106i	0.278664	−	0.482660i
$773$	4.89898	+	8.48528i	0.176204	+	0.305194i	0.940577	−	0.339580i	$-0.110285\pi$
$773$	4.89898	+	8.48528i	0.176204	+	0.305194i	−0.764373	+	0.644774i	$0.776951\pi$
$774$		0			0
$775$	−4.86396	−	2.80821i	−0.174719	−	0.100874i
$776$	6.63103			0.238040
$777$		0			0
$778$	32.4853			1.16465
$779$	10.3923	+	6.00000i	0.372343	+	0.214972i
$780$		0			0
$781$	27.0000	+	46.7654i	0.966136	+	1.67340i
$782$	−7.34847	+	12.7279i	−0.262781	+	0.455150i
$783$		0			0
$784$	6.74264	−	1.88064i	0.240809	−	0.0671656i
$785$			25.4558i			0.908558i
$786$		0			0
$787$	9.47056	−	5.46783i	0.337589	−	0.194907i	−0.321616	−	0.946870i	$-0.604226\pi$
$787$	9.47056	−	5.46783i	0.337589	−	0.194907i	0.659205	+	0.751963i	$0.270893\pi$
$788$	14.6969	−	8.48528i	0.523557	−	0.302276i
$789$		0			0
$790$		−	22.6398i		−	0.805486i
$791$	−4.30463	−	1.75736i	−0.153055	−	0.0624845i
$792$		0			0
$793$	−1.50000	+	2.59808i	−0.0532666	+	0.0922604i
$794$	14.9326	+	25.8640i	0.529937	+	0.917878i
$795$		0			0
$796$	3.10660	+	1.79360i	0.110111	+	0.0635724i
$797$	3.04384			0.107818			0.0539091	−	0.998546i	$-0.482832\pi$
$797$	3.04384			0.107818			0.0539091	+	0.998546i	$0.482832\pi$
$798$		0			0
$799$	−19.4558			−0.688298
$800$	−0.866025	−	0.500000i	−0.0306186	−	0.0176777i
$801$		0			0
$802$	−14.1213	−	24.4588i	−0.498641	−	0.863672i
$803$	−32.4377	+	56.1838i	−1.14470	+	1.98268i
$804$		0			0
$805$	−5.27208	−	38.5254i	−0.185816	−	1.35784i
$806$			23.4853i			0.827234i
$807$		0			0
$808$	6.36396	−	3.67423i	0.223883	−	0.129259i
$809$	2.15232	−	1.24264i	0.0756714	−	0.0436889i	−0.461687	−	0.887043i	$-0.652756\pi$
$809$	2.15232	−	1.24264i	0.0756714	−	0.0436889i	0.537358	+	0.843354i	$0.319422\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	3.67423	+	26.8492i	0.128940	+	0.942224i
$813$		0			0
$814$	−6.87868	+	11.9142i	−0.241098	+	0.417593i
$815$	9.16756	+	15.8787i	0.321126	+	0.556206i
$816$		0			0
$817$	29.6985	+	17.1464i	1.03902	+	0.599878i
$818$	13.5592			0.474087
$819$		0			0
$820$	6.00000			0.209529
$821$	9.50079	+	5.48528i	0.331580	+	0.191438i	0.656542	−	0.754289i	$-0.272018\pi$
$821$	9.50079	+	5.48528i	0.331580	+	0.191438i	−0.324963	+	0.945727i	$0.605352\pi$
$822$		0			0
$823$	5.62132	+	9.73641i	0.195947	+	0.339390i	0.947211	−	0.320612i	$-0.103889\pi$
$823$	5.62132	+	9.73641i	0.195947	+	0.339390i	−0.751264	+	0.660002i	$0.770555\pi$
$824$	3.10538	−	5.37868i	0.108181	−	0.187375i
$825$		0			0
$826$	−6.00000	−	2.44949i	−0.208767	−	0.0852286i
$827$		−	20.4853i		−	0.712343i	−0.934421	−	0.356172i	$-0.884082\pi$
$827$		−	20.4853i		−	0.712343i	0.934421	−	0.356172i	$-0.115918\pi$
$828$		0			0
$829$	−8.48528	+	4.89898i	−0.294706	+	0.170149i	−0.640062	−	0.768323i	$-0.721091\pi$
$829$	−8.48528	+	4.89898i	−0.294706	+	0.170149i	0.345356	+	0.938472i	$0.387758\pi$
$830$	−11.6531	+	6.72792i	−0.404485	+	0.233530i
$831$		0			0
$832$			4.18154i			0.144969i
$833$	4.26858	−	16.6066i	0.147898	−	0.575385i
$834$		0			0
$835$	−24.7279	+	42.8300i	−0.855745	+	1.48219i
$836$	10.3923	+	18.0000i	0.359425	+	0.622543i
$837$		0			0
$838$	11.1213	+	6.42090i	0.384180	+	0.221806i
$839$	−5.49333			−0.189651			−0.0948253	−	0.995494i	$-0.530229\pi$
$839$	−5.49333			−0.189651			−0.0948253	+	0.995494i	$0.530229\pi$
$840$		0			0
$841$	−75.9117			−2.61764
$842$	−23.7775	−	13.7279i	−0.819425	−	0.473095i
$843$		0			0
$844$	−6.74264	−	11.6786i	−0.232091	−	0.401994i
$845$	−5.49333	+	9.51472i	−0.188976	+	0.327316i
$846$		0			0
$847$	11.3492	+	14.6354i	0.389965	+	0.502878i
$848$		−	2.48528i		−	0.0853449i
$849$		0			0
$850$	−2.12132	+	1.22474i	−0.0727607	+	0.0420084i
$851$	−16.8493	+	9.72792i	−0.577585	+	0.333469i
$852$		0			0
$853$		−	51.3672i		−	1.75878i	−0.476103	−	0.879389i	$-0.657951\pi$
$853$		−	51.3672i		−	1.75878i	0.476103	−	0.879389i	$-0.342049\pi$
$854$	−1.88064	+	0.257359i	−0.0643541	+	0.00880665i
$855$		0			0
$856$	−7.24264	+	12.5446i	−0.247548	+	0.428766i
$857$	3.37706	+	5.84924i	0.115358	+	0.199806i	0.917923	−	0.396759i	$-0.129865\pi$
$857$	3.37706	+	5.84924i	0.115358	+	0.199806i	−0.802565	+	0.596565i	$0.796532\pi$
$858$		0			0
$859$	−9.47056	−	5.46783i	−0.323131	−	0.186560i	0.329656	−	0.944101i	$-0.393067\pi$
$859$	−9.47056	−	5.46783i	−0.323131	−	0.186560i	−0.652787	+	0.757541i	$0.726401\pi$
$860$	17.1464			0.584688
$861$		0			0
$862$	10.2426			0.348866
$863$	11.6531	+	6.72792i	0.396676	+	0.229021i	0.685049	−	0.728497i	$-0.259781\pi$
$863$	11.6531	+	6.72792i	0.396676	+	0.229021i	−0.288373	+	0.957518i	$0.593114\pi$
$864$		0			0
$865$	25.4558	+	44.0908i	0.865525	+	1.49913i
$866$	13.4106	−	23.2279i	0.455712	−	0.789317i
$867$		0			0
$868$	−11.7426	+	9.10601i	−0.398571	+	0.309078i
$869$			39.2132i			1.33022i
$870$		0			0
$871$	12.6213	−	7.28692i	0.427657	−	0.246908i
$872$	−6.69258	+	3.86396i	−0.226639	+	0.130850i
$873$		0			0
$874$			29.3939i			0.994263i
$875$	−9.79796	+	24.0000i	−0.331231	+	0.811348i
$876$		0			0
$877$	1.89340	−	3.27946i	0.0639355	−	0.110740i	−0.832286	−	0.554347i	$-0.812968\pi$
$877$	1.89340	−	3.27946i	0.0639355	−	0.110740i	0.896221	+	0.443607i	$0.146301\pi$
$878$	12.5446	+	21.7279i	0.423360	+	0.733282i
$879$		0			0
$880$	9.00000	+	5.19615i	0.303390	+	0.175162i
$881$	−4.30463			−0.145027			−0.0725134	−	0.997367i	$-0.523102\pi$
$881$	−4.30463			−0.145027			−0.0725134	+	0.997367i	$0.523102\pi$
$882$		0			0
$883$	−2.00000			−0.0673054			−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000			−0.0673054			−0.0336527	+	0.999434i	$0.510714\pi$
$884$	8.87039	+	5.12132i	0.298343	+	0.172249i
$885$		0			0
$886$	12.7279	+	22.0454i	0.427603	+	0.740630i
$887$	−14.9941	+	25.9706i	−0.503453	+	0.872006i	0.496539	+	0.868014i	$0.334604\pi$
$887$	−14.9941	+	25.9706i	−0.503453	+	0.872006i	−0.999992	+	0.00399177i	$0.998729\pi$
$888$		0			0
$889$	−17.7279	+	43.4244i	−0.594575	+	1.45641i
$890$		−	43.4558i		−	1.45664i
$891$		0			0
$892$	−9.00000	+	5.19615i	−0.301342	+	0.173980i
$893$	−33.6985	+	19.4558i	−1.12768	+	0.651065i
$894$		0			0
$895$		−	45.8739i		−	1.53339i
$896$	−2.09077	+	1.62132i	−0.0698477	+	0.0541645i
$897$		0			0
$898$	15.3640	−	26.6112i	0.512702	−	0.888026i
$899$	−28.7635	−	49.8198i	−0.959316	−	1.66158i
$900$		0			0
$901$	−5.27208	−	3.04384i	−0.175638	−	0.101405i
$902$	−10.3923			−0.346026
$903$		0			0
$904$	1.75736			0.0584489
$905$	−20.7846	−	12.0000i	−0.690904	−	0.398893i
$906$		0			0
$907$	−1.74264	−	3.01834i	−0.0578634	−	0.100222i	0.835643	−	0.549273i	$-0.185095\pi$
$907$	−1.74264	−	3.01834i	−0.0578634	−	0.100222i	−0.893506	+	0.449051i	$0.851762\pi$
$908$	2.15232	−	3.72792i	0.0714271	−	0.123715i
$909$		0			0
$910$	−26.8492	+	3.67423i	−0.890044	+	0.121800i
$911$		−	1.75736i		−	0.0582239i	−0.999576	−	0.0291120i	$-0.990732\pi$
$911$		−	1.75736i		−	0.0582239i	0.999576	−	0.0291120i	$-0.00926793\pi$
$912$		0			0
$913$	20.1838	−	11.6531i	0.667985	−	0.385661i
$914$	−19.9186	+	11.5000i	−0.658848	+	0.380386i
$915$		0			0
$916$			9.08052i			0.300029i
$917$		0			0
$918$		0			0
$919$	−0.136039	+	0.235626i	−0.00448751	+	0.00777260i	−0.868260	−	0.496109i	$-0.834762\pi$
$919$	−0.136039	+	0.235626i	−0.00448751	+	0.00777260i	0.863773	+	0.503881i	$0.168095\pi$
$920$	7.34847	+	12.7279i	0.242272	+	0.419627i
$921$		0			0
$922$	−36.5772	−	21.1178i	−1.20460	−	0.695479i
$923$	53.2223			1.75183
$924$		0			0
$925$	−3.24264			−0.106617
$926$	19.0526	+	11.0000i	0.626106	+	0.361482i
$927$		0			0
$928$	−5.12132	−	8.87039i	−0.168116	−	0.291185i
$929$	16.8132	−	29.1213i	0.551623	−	0.955440i	−0.446534	−	0.894766i	$-0.647342\pi$
$929$	16.8132	−	29.1213i	0.551623	−	0.955440i	0.998158	−	0.0606731i	$-0.0193247\pi$
$930$		0			0
$931$	−9.21320	−	33.0321i	−0.301951	−	1.08258i
$932$		−	3.51472i		−	0.115128i
$933$		0			0
$934$	−2.63604	+	1.52192i	−0.0862538	+	0.0497987i
$935$	22.0454	−	12.7279i	0.720962	−	0.416248i
$936$		0			0
$937$			53.0992i			1.73468i	0.497719	+	0.867338i	$0.334171\pi$
$937$			53.0992i			1.73468i	−0.497719	+	0.867338i	$0.665829\pi$
$938$	8.53716	+	3.48528i	0.278748	+	0.113798i
$939$		0			0
$940$	−9.72792	+	16.8493i	−0.317290	+	0.549562i
$941$	−10.7255	−	18.5772i	−0.349642	−	0.605598i	0.636544	−	0.771241i	$-0.280364\pi$
$941$	−10.7255	−	18.5772i	−0.349642	−	0.605598i	−0.986186	+	0.165643i	$0.947030\pi$
$942$		0			0
$943$	−12.7279	−	7.34847i	−0.414478	−	0.239299i
$944$	2.44949			0.0797241
$945$		0			0
$946$	−29.6985			−0.965581
$947$	6.71807	+	3.87868i	0.218308	+	0.126040i	0.605167	−	0.796099i	$-0.293107\pi$
$947$	6.71807	+	3.87868i	0.218308	+	0.126040i	−0.386859	+	0.922139i	$0.626440\pi$
$948$		0			0
$949$	31.9706	+	55.3746i	1.03781	+	1.79754i
$950$	−2.44949	+	4.24264i	−0.0794719	+	0.137649i
$951$		0			0
$952$	0.878680	+	6.42090i	0.0284782	+	0.208102i
$953$		−	34.9706i		−	1.13281i	−0.824128	−	0.566404i	$-0.808334\pi$
$953$		−	34.9706i		−	1.13281i	0.824128	−	0.566404i	$-0.191666\pi$
$954$		0			0
$955$	45.0000	−	25.9808i	1.45617	−	0.840718i
$956$	−6.71807	+	3.87868i	−0.217278	+	0.125445i
$957$		0			0
$958$			2.44949i			0.0791394i
$959$	2.15232	+	15.7279i	0.0695019	+	0.507881i
$960$		0			0
$961$	0.272078	−	0.471253i	0.00877671	−	0.0152017i
$962$	6.77962	+	11.7426i	0.218584	+	0.378598i
$963$		0			0
$964$	15.9853	+	9.22911i	0.514851	+	0.297250i
$965$	−37.9310			−1.22104
$966$		0			0
$967$	16.6985			0.536987			0.268494	−	0.963281i	$-0.413474\pi$
$967$	16.6985			0.536987			0.268494	+	0.963281i	$0.413474\pi$
$968$	−6.06218	−	3.50000i	−0.194846	−	0.112494i
$969$		0			0
$970$	−8.12132	−	14.0665i	−0.260760	−	0.451649i
$971$	−0.594346	+	1.02944i	−0.0190735	+	0.0330362i	−0.875405	−	0.483391i	$-0.839405\pi$
$971$	−0.594346	+	1.02944i	−0.0190735	+	0.0330362i	0.856331	+	0.516427i	$0.172738\pi$
$972$		0			0
$973$	−28.2426	−	11.5300i	−0.905417	−	0.369635i
$974$			22.0000i			0.704925i
$975$		0			0
$976$	0.621320	−	0.358719i	0.0198880	−	0.0114823i
$977$	5.19615	−	3.00000i	0.166240	−	0.0959785i	−0.414572	−	0.910017i	$-0.636069\pi$
$977$	5.19615	−	3.00000i	0.166240	−	0.0959785i	0.580812	+	0.814038i	$0.302735\pi$
$978$		0			0
$979$			75.2677i			2.40557i
$980$	−12.2474	−	12.0000i	−0.391230	−	0.383326i
$981$		0			0
$982$	0.514719	−	0.891519i	0.0164253	−	0.0284495i
$983$	9.16756	+	15.8787i	0.292400	+	0.506451i	0.974377	−	0.224922i	$-0.0722129\pi$
$983$	9.16756	+	15.8787i	0.292400	+	0.506451i	−0.681977	+	0.731374i	$0.738880\pi$
$984$		0			0
$985$	−36.0000	−	20.7846i	−1.14706	−	0.662253i
$986$	−25.0892			−0.799004
$987$		0			0
$988$	20.4853			0.651724
$989$	−36.3731	−	21.0000i	−1.15660	−	0.667761i
$990$		0			0
$991$	−11.1066	−	19.2372i	−0.352813	−	0.611090i	0.633928	−	0.773392i	$-0.281441\pi$
$991$	−11.1066	−	19.2372i	−0.352813	−	0.611090i	−0.986741	+	0.162302i	$0.948108\pi$
$992$	2.80821	−	4.86396i	0.0891607	−	0.154431i
$993$		0			0
$994$	20.6360	+	26.6112i	0.654535	+	0.844055i
$995$		−	8.78680i		−	0.278560i
$996$		0			0
$997$	4.13604	−	2.38794i	0.130990	−	0.0756269i	−0.433073	−	0.901359i	$-0.642571\pi$
$997$	4.13604	−	2.38794i	0.130990	−	0.0756269i	0.564063	+	0.825732i	$0.309238\pi$
$998$	22.9114	−	13.2279i	0.725249	−	0.418723i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.k.d.215.2	✓	8
3.2	odd	2	inner	378.2.k.d.215.3	yes	8
7.2	even	3		2646.2.d.d.2645.6		8
7.3	odd	6	inner	378.2.k.d.269.3	yes	8
7.5	odd	6		2646.2.d.d.2645.8		8
9.2	odd	6		1134.2.l.e.215.1		8
9.4	even	3		1134.2.t.f.593.3		8
9.5	odd	6		1134.2.t.f.593.2		8
9.7	even	3		1134.2.l.e.215.4		8
21.2	odd	6		2646.2.d.d.2645.3		8
21.5	even	6		2646.2.d.d.2645.1		8
21.17	even	6	inner	378.2.k.d.269.2	yes	8
63.31	odd	6		1134.2.l.e.269.3		8
63.38	even	6		1134.2.t.f.1025.3		8
63.52	odd	6		1134.2.t.f.1025.2		8
63.59	even	6		1134.2.l.e.269.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.k.d.215.2	✓	8	1.1	even	1	trivial
378.2.k.d.215.3	yes	8	3.2	odd	2	inner
378.2.k.d.269.2	yes	8	21.17	even	6	inner
378.2.k.d.269.3	yes	8	7.3	odd	6	inner
1134.2.l.e.215.1		8	9.2	odd	6
1134.2.l.e.215.4		8	9.7	even	3
1134.2.l.e.269.2		8	63.59	even	6
1134.2.l.e.269.3		8	63.31	odd	6
1134.2.t.f.593.2		8	9.5	odd	6
1134.2.t.f.593.3		8	9.4	even	3
1134.2.t.f.1025.2		8	63.52	odd	6
1134.2.t.f.1025.3		8	63.38	even	6
2646.2.d.d.2645.1		8	21.5	even	6
2646.2.d.d.2645.3		8	21.2	odd	6
2646.2.d.d.2645.6		8	7.2	even	3
2646.2.d.d.2645.8		8	7.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.22474 - 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.22474 - 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +(-2.12132 + 1.22474i) q^{10} +(3.67423 - 2.12132i) q^{11} -4.18154i q^{13} +(2.09077 - 1.62132i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.22474 + 2.12132i) q^{17} +(4.24264 + 2.44949i) q^{19} +2.44949 q^{20} -4.24264 q^{22} +(-5.19615 - 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.09077 + 3.62132i) q^{26} +(-2.62132 + 0.358719i) q^{28} -10.2426i q^{29} +(4.86396 - 2.80821i) q^{31} +(0.866025 - 0.500000i) q^{32} -2.44949i q^{34} +(3.97141 + 5.12132i) q^{35} +(1.62132 - 2.80821i) q^{37} +(-2.44949 - 4.24264i) q^{38} +(-2.12132 - 1.22474i) q^{40} +2.44949 q^{41} +7.00000 q^{43} +(3.67423 + 2.12132i) q^{44} +(3.00000 + 5.19615i) q^{46} +(-3.97141 + 6.87868i) q^{47} +(-5.00000 - 4.89898i) q^{49} +1.00000i q^{50} +(3.62132 - 2.09077i) q^{52} +(-2.15232 + 1.24264i) q^{53} -10.3923i q^{55} +(2.44949 + 1.00000i) q^{56} +(-5.12132 + 8.87039i) q^{58} +(-1.22474 - 2.12132i) q^{59} +(-0.621320 - 0.358719i) q^{61} -5.61642 q^{62} -1.00000 q^{64} +(-8.87039 - 5.12132i) q^{65} +(1.74264 + 3.01834i) q^{67} +(-1.22474 + 2.12132i) q^{68} +(-0.878680 - 6.42090i) q^{70} +12.7279i q^{71} +(-13.2426 + 7.64564i) q^{73} +(-2.80821 + 1.62132i) q^{74} +4.89898i q^{76} +(1.52192 + 11.1213i) q^{77} +(-4.62132 + 8.00436i) q^{79} +(1.22474 + 2.12132i) q^{80} +(-2.12132 - 1.22474i) q^{82} +5.49333 q^{83} +6.00000 q^{85} +(-6.06218 - 3.50000i) q^{86} +(-2.12132 - 3.67423i) q^{88} +(-8.87039 + 15.3640i) q^{89} +(10.2426 + 4.18154i) q^{91} -6.00000i q^{92} +(6.87868 - 3.97141i) q^{94} +(10.3923 - 6.00000i) q^{95} +6.63103i q^{97} +(1.88064 + 6.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q + 4 * q^4 - 8 * q^7	\( 8 q + 4 q^{4} - 8 q^{7} - 4 q^{16} - 4 q^{25} - 4 q^{28} - 12 q^{31} - 4 q^{37} + 56 q^{43} + 24 q^{46} - 40 q^{49} + 12 q^{52} - 24 q^{58} + 12 q^{61} - 8 q^{64} - 20 q^{67} - 24 q^{70} - 72 q^{73} - 20 q^{79} + 48 q^{85} + 48 q^{91} + 72 q^{94}+O(q^{100}) \) 8 * q + 4 * q^4 - 8 * q^7 - 4 * q^16 - 4 * q^25 - 4 * q^28 - 12 * q^31 - 4 * q^37 + 56 * q^43 + 24 * q^46 - 40 * q^49 + 12 * q^52 - 24 * q^58 + 12 * q^61 - 8 * q^64 - 20 * q^67 - 24 * q^70 - 72 * q^73 - 20 * q^79 + 48 * q^85 + 48 * q^91 + 72 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more