LMFDB - Embedded newform 784.2.x.e.165.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(165,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 8]))

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

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

chi := DirichletCharacter("784.165");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.x (of order $12$, degree $4$, 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:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			165.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	784.165
Dual form			784.2.x.e.765.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(-0.732051 + 2.73205i) q^{5} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})$	\(q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(-0.732051 + 2.73205i) q^{5} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +(-2.00000 + 3.46410i) q^{10} +(-1.36603 + 0.366025i) q^{11} +(2.00000 + 3.46410i) q^{16} +(-1.00000 + 1.73205i) q^{17} +(4.09808 - 1.09808i) q^{18} +(2.73205 + 0.732051i) q^{19} +(-4.00000 + 4.00000i) q^{20} -2.00000 q^{22} +(-5.19615 + 3.00000i) q^{23} +(-2.59808 - 1.50000i) q^{25} +(7.00000 - 7.00000i) q^{29} +(-4.00000 + 6.92820i) q^{31} +(1.46410 + 5.46410i) q^{32} +(-2.00000 + 2.00000i) q^{34} +6.00000 q^{36} +(-1.83013 + 6.83013i) q^{37} +(3.46410 + 2.00000i) q^{38} +(-6.92820 + 4.00000i) q^{40} -10.0000i q^{41} +(-1.00000 - 1.00000i) q^{43} +(-2.73205 - 0.732051i) q^{44} +(2.19615 + 8.19615i) q^{45} +(-8.19615 + 2.19615i) q^{46} +(-6.00000 - 10.3923i) q^{47} +(-3.00000 - 3.00000i) q^{50} +(1.36603 - 0.366025i) q^{53} -4.00000i q^{55} +(12.1244 - 7.00000i) q^{58} +(10.9282 - 2.92820i) q^{59} +(8.19615 + 2.19615i) q^{61} +(-8.00000 + 8.00000i) q^{62} +8.00000i q^{64} +(1.09808 + 4.09808i) q^{67} +(-3.46410 + 2.00000i) q^{68} +(8.19615 + 2.19615i) q^{72} +(-5.19615 - 3.00000i) q^{73} +(-5.00000 + 8.66025i) q^{74} +(4.00000 + 4.00000i) q^{76} +(-5.00000 - 8.66025i) q^{79} +(-10.9282 + 2.92820i) q^{80} +(4.50000 - 7.79423i) q^{81} +(3.66025 - 13.6603i) q^{82} +(10.0000 - 10.0000i) q^{83} +(-4.00000 - 4.00000i) q^{85} +(-1.00000 - 1.73205i) q^{86} +(-3.46410 - 2.00000i) q^{88} +(12.1244 - 7.00000i) q^{89} +12.0000i q^{90} -12.0000 q^{92} +(-4.39230 - 16.3923i) q^{94} +(-4.00000 + 6.92820i) q^{95} +2.00000 q^{97} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{5} + 8 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^5 + 8 * q^8	$ 4 q + 2 q^{2} + 4 q^{5} + 8 q^{8} - 8 q^{10} - 2 q^{11} + 8 q^{16} - 4 q^{17} + 6 q^{18} + 4 q^{19} - 16 q^{20} - 8 q^{22} + 28 q^{29} - 16 q^{31} - 8 q^{32} - 8 q^{34} + 24 q^{36} + 10 q^{37} - 4 q^{43} - 4 q^{44} - 12 q^{45} - 12 q^{46} - 24 q^{47} - 12 q^{50} + 2 q^{53} + 16 q^{59} + 12 q^{61} - 32 q^{62} - 6 q^{67} + 12 q^{72} - 20 q^{74} + 16 q^{76} - 20 q^{79} - 16 q^{80} + 18 q^{81} - 20 q^{82} + 40 q^{83} - 16 q^{85} - 4 q^{86} - 48 q^{92} + 24 q^{94} - 16 q^{95} + 8 q^{97} - 12 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^5 + 8 * q^8 - 8 * q^10 - 2 * q^11 + 8 * q^16 - 4 * q^17 + 6 * q^18 + 4 * q^19 - 16 * q^20 - 8 * q^22 + 28 * q^29 - 16 * q^31 - 8 * q^32 - 8 * q^34 + 24 * q^36 + 10 * q^37 - 4 * q^43 - 4 * q^44 - 12 * q^45 - 12 * q^46 - 24 * q^47 - 12 * q^50 + 2 * q^53 + 16 * q^59 + 12 * q^61 - 32 * q^62 - 6 * q^67 + 12 * q^72 - 20 * q^74 + 16 * q^76 - 20 * q^79 - 16 * q^80 + 18 * q^81 - 20 * q^82 + 40 * q^83 - 16 * q^85 - 4 * q^86 - 48 * q^92 + 24 * q^94 - 16 * q^95 + 8 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.36603	+	0.366025i	0.965926	+	0.258819i
$2$	1.36603	+	0.366025i	0.965926	+	0.258819i
$3$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$3$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$4$	1.73205	+	1.00000i	0.866025	+	0.500000i
$5$	−0.732051	+	2.73205i	−0.327383	+	1.22181i	0.584511	+	0.811386i	$0.301286\pi$
$5$	−0.732051	+	2.73205i	−0.327383	+	1.22181i	−0.911894	+	0.410425i	$0.865380\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$	2.59808	−	1.50000i	0.866025	−	0.500000i
$10$	−2.00000	+	3.46410i	−0.632456	+	1.09545i
$11$	−1.36603	+	0.366025i	−0.411872	+	0.110361i	−0.458804	−	0.888537i	$-0.651722\pi$
$11$	−1.36603	+	0.366025i	−0.411872	+	0.110361i	0.0469323	+	0.998898i	$0.485055\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$	4.09808	−	1.09808i	0.965926	−	0.258819i
$19$	2.73205	+	0.732051i	0.626775	+	0.167944i	0.558206	−	0.829702i	$-0.311490\pi$
$19$	2.73205	+	0.732051i	0.626775	+	0.167944i	0.0685694	+	0.997646i	$0.478157\pi$
$20$	−4.00000	+	4.00000i	−0.894427	+	0.894427i
$21$		0			0
$22$	−2.00000			−0.426401
$23$	−5.19615	+	3.00000i	−1.08347	+	0.625543i	−0.931831	−	0.362892i	$-0.881789\pi$
$23$	−5.19615	+	3.00000i	−1.08347	+	0.625543i	−0.151642	+	0.988436i	$0.548456\pi$
$24$		0			0
$25$	−2.59808	−	1.50000i	−0.519615	−	0.300000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.00000	−	7.00000i	1.29987	−	1.29987i	0.371391	−	0.928477i	$-0.378881\pi$
$29$	7.00000	−	7.00000i	1.29987	−	1.29987i	0.928477	−	0.371391i	$-0.121119\pi$
$30$		0			0
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	0.243204	+	0.969975i	$0.421802\pi$
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	−0.961625	+	0.274367i	$0.911532\pi$
$32$	1.46410	+	5.46410i	0.258819	+	0.965926i
$33$		0			0
$34$	−2.00000	+	2.00000i	−0.342997	+	0.342997i
$35$		0			0
$36$	6.00000			1.00000
$37$	−1.83013	+	6.83013i	−0.300871	+	1.12287i	0.635571	+	0.772043i	$0.280765\pi$
$37$	−1.83013	+	6.83013i	−0.300871	+	1.12287i	−0.936442	+	0.350823i	$0.885902\pi$
$38$	3.46410	+	2.00000i	0.561951	+	0.324443i
$39$		0			0
$40$	−6.92820	+	4.00000i	−1.09545	+	0.632456i
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$	−1.00000	−	1.00000i	−0.152499	−	0.152499i	0.626734	−	0.779233i	$-0.284391\pi$
$43$	−1.00000	−	1.00000i	−0.152499	−	0.152499i	−0.779233	+	0.626734i	$0.784391\pi$
$44$	−2.73205	−	0.732051i	−0.411872	−	0.110361i
$45$	2.19615	+	8.19615i	0.327383	+	1.22181i
$46$	−8.19615	+	2.19615i	−1.20846	+	0.323805i
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.856560	−	0.516047i	$-0.827403\pi$
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.0186297	−	0.999826i	$-0.505930\pi$
$48$		0			0
$49$		0			0
$50$	−3.00000	−	3.00000i	−0.424264	−	0.424264i
$51$		0			0
$52$		0			0
$53$	1.36603	−	0.366025i	0.187638	−	0.0502775i	−0.163776	−	0.986498i	$-0.552368\pi$
$53$	1.36603	−	0.366025i	0.187638	−	0.0502775i	0.351414	+	0.936220i	$0.385701\pi$
$54$		0			0
$55$		−	4.00000i		−	0.539360i
$56$		0			0
$57$		0			0
$58$	12.1244	−	7.00000i	1.59201	−	0.919145i
$59$	10.9282	−	2.92820i	1.42273	−	0.381220i	0.536279	−	0.844041i	$-0.319829\pi$
$59$	10.9282	−	2.92820i	1.42273	−	0.381220i	0.886452	+	0.462821i	$0.153163\pi$
$60$		0			0
$61$	8.19615	+	2.19615i	1.04941	+	0.281189i	0.742008	−	0.670391i	$-0.233874\pi$
$61$	8.19615	+	2.19615i	1.04941	+	0.281189i	0.307402	+	0.951580i	$0.400540\pi$
$62$	−8.00000	+	8.00000i	−1.01600	+	1.01600i
$63$		0			0
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	1.09808	+	4.09808i	0.134151	+	0.500660i	1.00000		0.000353546i	$0.000112537\pi$
$67$	1.09808	+	4.09808i	0.134151	+	0.500660i	−0.865849	+	0.500306i	$0.833221\pi$
$68$	−3.46410	+	2.00000i	−0.420084	+	0.242536i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	8.19615	+	2.19615i	0.965926	+	0.258819i
$73$	−5.19615	−	3.00000i	−0.608164	−	0.351123i	0.164083	−	0.986447i	$-0.447534\pi$
$73$	−5.19615	−	3.00000i	−0.608164	−	0.351123i	−0.772246	+	0.635323i	$0.780867\pi$
$74$	−5.00000	+	8.66025i	−0.581238	+	1.00673i
$75$		0			0
$76$	4.00000	+	4.00000i	0.458831	+	0.458831i
$77$		0			0
$78$		0			0
$79$	−5.00000	−	8.66025i	−0.562544	−	0.974355i	−0.997274	−	0.0737937i	$-0.976489\pi$
$79$	−5.00000	−	8.66025i	−0.562544	−	0.974355i	0.434730	−	0.900561i	$-0.356844\pi$
$80$	−10.9282	+	2.92820i	−1.22181	+	0.327383i
$81$	4.50000	−	7.79423i	0.500000	−	0.866025i
$82$	3.66025	−	13.6603i	0.404207	−	1.50852i
$83$	10.0000	−	10.0000i	1.09764	−	1.09764i	0.102957	−	0.994686i	$-0.467170\pi$
$83$	10.0000	−	10.0000i	1.09764	−	1.09764i	0.994686	−	0.102957i	$-0.0328303\pi$
$84$		0			0
$85$	−4.00000	−	4.00000i	−0.433861	−	0.433861i
$86$	−1.00000	−	1.73205i	−0.107833	−	0.186772i
$87$		0			0
$88$	−3.46410	−	2.00000i	−0.369274	−	0.213201i
$89$	12.1244	−	7.00000i	1.28518	−	0.741999i	0.307389	−	0.951584i	$-0.400545\pi$
$89$	12.1244	−	7.00000i	1.28518	−	0.741999i	0.977790	+	0.209585i	$0.0672115\pi$
$90$			12.0000i			1.26491i
$91$		0			0
$92$	−12.0000			−1.25109
$93$		0			0
$94$	−4.39230	−	16.3923i	−0.453032	−	1.69074i
$95$	−4.00000	+	6.92820i	−0.410391	+	0.710819i
$96$		0			0
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$	−3.00000	+	3.00000i	−0.301511	+	0.301511i
$100$	−3.00000	−	5.19615i	−0.300000	−	0.519615i
$101$	8.19615	−	2.19615i	0.815548	−	0.218525i	0.173149	−	0.984896i	$-0.444606\pi$
$101$	8.19615	−	2.19615i	0.815548	−	0.218525i	0.642399	+	0.766370i	$0.277939\pi$
$102$		0			0
$103$	−3.46410	+	2.00000i	−0.341328	+	0.197066i	−0.660859	−	0.750510i	$-0.729808\pi$
$103$	−3.46410	+	2.00000i	−0.341328	+	0.197066i	0.319531	+	0.947576i	$0.396475\pi$
$104$		0			0
$105$		0			0
$106$	2.00000			0.194257
$107$	1.83013	−	6.83013i	0.176925	−	0.660293i	−0.819291	−	0.573378i	$-0.805633\pi$
$107$	1.83013	−	6.83013i	0.176925	−	0.660293i	0.996216	−	0.0869149i	$-0.0277008\pi$
$108$		0			0
$109$	−1.09808	−	4.09808i	−0.105177	−	0.392525i	0.893189	−	0.449682i	$-0.148463\pi$
$109$	−1.09808	−	4.09808i	−0.105177	−	0.392525i	−0.998365	+	0.0571579i	$0.981796\pi$
$110$	1.46410	−	5.46410i	0.139597	−	0.520982i
$111$		0			0
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	−4.39230	−	16.3923i	−0.409585	−	1.52859i
$116$	19.1244	−	5.12436i	1.77565	−	0.475784i
$117$		0			0
$118$	16.0000			1.47292
$119$		0			0
$120$		0			0
$121$	−7.79423	+	4.50000i	−0.708566	+	0.409091i
$122$	10.3923	+	6.00000i	0.940875	+	0.543214i
$123$		0			0
$124$	−13.8564	+	8.00000i	−1.24434	+	0.718421i
$125$	−4.00000	+	4.00000i	−0.357771	+	0.357771i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−2.92820	+	10.9282i	−0.258819	+	0.965926i
$129$		0			0
$130$		0			0
$131$	−19.1244	−	5.12436i	−1.67090	−	0.447717i	−0.705549	−	0.708661i	$-0.749299\pi$
$131$	−19.1244	−	5.12436i	−1.67090	−	0.447717i	−0.965354	+	0.260944i	$0.915966\pi$
$132$		0			0
$133$		0			0
$134$			6.00000i			0.518321i
$135$		0			0
$136$	−5.46410	+	1.46410i	−0.468543	+	0.125546i
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.0146279	−	0.999893i	$-0.504656\pi$
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.873247	+	0.487278i	$0.837990\pi$
$138$		0			0
$139$	2.00000	+	2.00000i	0.169638	+	0.169638i	0.786820	−	0.617182i	$-0.211726\pi$
$139$	2.00000	+	2.00000i	0.169638	+	0.169638i	−0.617182	+	0.786820i	$0.711726\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	10.3923	+	6.00000i	0.866025	+	0.500000i
$145$	14.0000	+	24.2487i	1.16264	+	2.01375i
$146$	−6.00000	−	6.00000i	−0.496564	−	0.496564i
$147$		0			0
$148$	−10.0000	+	10.0000i	−0.821995	+	0.821995i
$149$	1.09808	−	4.09808i	0.0899579	−	0.335727i	−0.906249	−	0.422744i	$-0.861067\pi$
$149$	1.09808	−	4.09808i	0.0899579	−	0.335727i	0.996207	+	0.0870170i	$0.0277334\pi$
$150$		0			0
$151$	−8.66025	−	5.00000i	−0.704761	−	0.406894i	0.104357	−	0.994540i	$-0.466722\pi$
$151$	−8.66025	−	5.00000i	−0.704761	−	0.406894i	−0.809118	+	0.587646i	$0.800055\pi$
$152$	4.00000	+	6.92820i	0.324443	+	0.561951i
$153$			6.00000i			0.485071i
$154$		0			0
$155$	−16.0000	−	16.0000i	−1.28515	−	1.28515i
$156$		0			0
$157$	4.39230	+	16.3923i	0.350544	+	1.30825i	0.886000	+	0.463685i	$0.153473\pi$
$157$	4.39230	+	16.3923i	0.350544	+	1.30825i	−0.535456	+	0.844563i	$0.679860\pi$
$158$	−3.66025	−	13.6603i	−0.291194	−	1.08675i
$159$		0			0
$160$	−16.0000			−1.26491
$161$		0			0
$162$	9.00000	−	9.00000i	0.707107	−	0.707107i
$163$	−20.4904	−	5.49038i	−1.60493	−	0.430040i	−0.658404	−	0.752665i	$-0.728768\pi$
$163$	−20.4904	−	5.49038i	−1.60493	−	0.430040i	−0.946527	+	0.322625i	$0.895435\pi$
$164$	10.0000	−	17.3205i	0.780869	−	1.35250i
$165$		0			0
$166$	17.3205	−	10.0000i	1.34433	−	0.776151i
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$	−4.00000	−	6.92820i	−0.306786	−	0.531369i
$171$	8.19615	−	2.19615i	0.626775	−	0.167944i
$172$	−0.732051	−	2.73205i	−0.0558184	−	0.208317i
$173$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$173$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$174$		0			0
$175$		0			0
$176$	−4.00000	−	4.00000i	−0.301511	−	0.301511i
$177$		0			0
$178$	19.1244	−	5.12436i	1.43343	−	0.384087i
$179$	−1.09808	−	4.09808i	−0.0820741	−	0.306305i	0.912670	−	0.408697i	$-0.134017\pi$
$179$	−1.09808	−	4.09808i	−0.0820741	−	0.306305i	−0.994744	+	0.102393i	$0.967350\pi$
$180$	−4.39230	+	16.3923i	−0.327383	+	1.22181i
$181$	4.00000	+	4.00000i	0.297318	+	0.297318i	0.839962	−	0.542645i	$-0.182577\pi$
$181$	4.00000	+	4.00000i	0.297318	+	0.297318i	−0.542645	+	0.839962i	$0.682577\pi$
$182$		0			0
$183$		0			0
$184$	−16.3923	−	4.39230i	−1.20846	−	0.323805i
$185$	−17.3205	−	10.0000i	−1.27343	−	0.735215i
$186$		0			0
$187$	0.732051	−	2.73205i	0.0535329	−	0.199787i
$188$		−	24.0000i		−	1.75038i
$189$		0			0
$190$	−8.00000	+	8.00000i	−0.580381	+	0.580381i
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	0.982828	+	0.184525i	$0.0590746\pi$
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	−0.331611	+	0.943416i	$0.607592\pi$
$192$		0			0
$193$	8.00000	−	13.8564i	0.575853	−	0.997406i	−0.420096	−	0.907480i	$-0.638004\pi$
$193$	8.00000	−	13.8564i	0.575853	−	0.997406i	0.995948	−	0.0899262i	$-0.0286631\pi$
$194$	2.73205	+	0.732051i	0.196150	+	0.0525582i
$195$		0			0
$196$		0			0
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	0.862423	−	0.506188i	$-0.168946\pi$
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	−0.506188	+	0.862423i	$0.668946\pi$
$198$	−5.19615	+	3.00000i	−0.369274	+	0.213201i
$199$	−20.7846	−	12.0000i	−1.47338	−	0.850657i	−0.473831	−	0.880616i	$-0.657129\pi$
$199$	−20.7846	−	12.0000i	−1.47338	−	0.850657i	−0.999551	+	0.0299585i	$0.990462\pi$
$200$	−2.19615	−	8.19615i	−0.155291	−	0.579555i
$201$		0			0
$202$	12.0000			0.844317
$203$		0			0
$204$		0			0
$205$	27.3205	+	7.32051i	1.90815	+	0.511286i
$206$	−5.46410	+	1.46410i	−0.380702	+	0.102009i
$207$	−9.00000	+	15.5885i	−0.625543	+	1.08347i
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	−0.945425	−	0.325840i	$-0.894353\pi$
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	0.325840	+	0.945425i	$0.394353\pi$
$212$	2.73205	+	0.732051i	0.187638	+	0.0502775i
$213$		0			0
$214$	5.00000	−	8.66025i	0.341793	−	0.592003i
$215$	3.46410	−	2.00000i	0.236250	−	0.136399i
$216$		0			0
$217$		0			0
$218$		−	6.00000i		−	0.406371i
$219$		0			0
$220$	4.00000	−	6.92820i	0.269680	−	0.467099i
$221$		0			0
$222$		0			0
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$		0			0
$225$	−9.00000			−0.600000
$226$	5.46410	+	1.46410i	0.363467	+	0.0973906i
$227$	0.732051	+	2.73205i	0.0485879	+	0.181333i	0.985955	−	0.167010i	$-0.0534112\pi$
$227$	0.732051	+	2.73205i	0.0485879	+	0.181333i	−0.937367	+	0.348343i	$0.886745\pi$
$228$		0			0
$229$	−2.92820	+	10.9282i	−0.193501	+	0.722156i	0.799149	+	0.601133i	$0.205284\pi$
$229$	−2.92820	+	10.9282i	−0.193501	+	0.722156i	−0.992650	+	0.121023i	$0.961383\pi$
$230$		−	24.0000i		−	1.58251i
$231$		0			0
$232$	28.0000			1.83829
$233$	−13.8564	+	8.00000i	−0.907763	+	0.524097i	−0.879711	−	0.475509i	$-0.842264\pi$
$233$	−13.8564	+	8.00000i	−0.907763	+	0.524097i	−0.0280525	+	0.999606i	$0.508931\pi$
$234$		0			0
$235$	32.7846	−	8.78461i	2.13863	−	0.573045i
$236$	21.8564	+	5.85641i	1.42273	+	0.381220i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	11.0000	−	19.0526i	0.708572	−	1.22728i	−0.256814	−	0.966461i	$-0.582673\pi$
$241$	11.0000	−	19.0526i	0.708572	−	1.22728i	0.965387	−	0.260822i	$-0.0839937\pi$
$242$	−12.2942	+	3.29423i	−0.790303	+	0.211761i
$243$		0			0
$244$	12.0000	+	12.0000i	0.768221	+	0.768221i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−21.8564	+	5.85641i	−1.38788	+	0.371882i
$249$		0			0
$250$	−6.92820	+	4.00000i	−0.438178	+	0.252982i
$251$	14.0000	+	14.0000i	0.883672	+	0.883672i	0.993906	−	0.110234i	$-0.0351599\pi$
$251$	14.0000	+	14.0000i	0.883672	+	0.883672i	−0.110234	+	0.993906i	$0.535160\pi$
$252$		0			0
$253$	6.00000	−	6.00000i	0.377217	−	0.377217i
$254$	10.9282	+	2.92820i	0.685696	+	0.183732i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−1.00000	−	1.73205i	−0.0623783	−	0.108042i	0.833150	−	0.553047i	$-0.186535\pi$
$257$	−1.00000	−	1.73205i	−0.0623783	−	0.108042i	−0.895528	+	0.445005i	$0.853202\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	7.68653	−	28.6865i	0.475784	−	1.77565i
$262$	−24.2487	−	14.0000i	−1.49809	−	0.864923i
$263$	13.8564	+	8.00000i	0.854423	+	0.493301i	0.862141	−	0.506669i	$-0.169123\pi$
$263$	13.8564	+	8.00000i	0.854423	+	0.493301i	−0.00771799	+	0.999970i	$0.502457\pi$
$264$		0			0
$265$			4.00000i			0.245718i
$266$		0			0
$267$		0			0
$268$	−2.19615	+	8.19615i	−0.134151	+	0.500660i
$269$	2.92820	+	10.9282i	0.178536	+	0.666304i	0.995922	+	0.0902148i	$0.0287554\pi$
$269$	2.92820	+	10.9282i	0.178536	+	0.666304i	−0.817387	+	0.576089i	$0.804578\pi$
$270$		0			0
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	0.718580	−	0.695444i	$-0.244792\pi$
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	−0.961563	+	0.274586i	$0.911459\pi$
$272$	−8.00000			−0.485071
$273$		0			0
$274$	−12.0000	−	12.0000i	−0.724947	−	0.724947i
$275$	4.09808	+	1.09808i	0.247123	+	0.0662165i
$276$		0			0
$277$	20.4904	−	5.49038i	1.23115	−	0.329885i	0.416121	−	0.909309i	$-0.363389\pi$
$277$	20.4904	−	5.49038i	1.23115	−	0.329885i	0.815026	+	0.579424i	$0.196722\pi$
$278$	2.00000	+	3.46410i	0.119952	+	0.207763i
$279$			24.0000i			1.43684i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	5.46410	−	1.46410i	0.324807	−	0.0870318i	−0.0927310	−	0.995691i	$-0.529560\pi$
$283$	5.46410	−	1.46410i	0.324807	−	0.0870318i	0.417538	+	0.908659i	$0.362893\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	12.0000	+	12.0000i	0.707107	+	0.707107i
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$	10.2487	+	38.2487i	0.601825	+	2.24604i
$291$		0			0
$292$	−6.00000	−	10.3923i	−0.351123	−	0.608164i
$293$	−14.0000	−	14.0000i	−0.817889	−	0.817889i	0.167913	−	0.985802i	$-0.446297\pi$
$293$	−14.0000	−	14.0000i	−0.817889	−	0.817889i	−0.985802	+	0.167913i	$0.946297\pi$
$294$		0			0
$295$			32.0000i			1.86311i
$296$	−17.3205	+	10.0000i	−1.00673	+	0.581238i
$297$		0			0
$298$	3.00000	−	5.19615i	0.173785	−	0.301005i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−10.0000	−	10.0000i	−0.575435	−	0.575435i
$303$		0			0
$304$	2.92820	+	10.9282i	0.167944	+	0.626775i
$305$	−12.0000	+	20.7846i	−0.687118	+	1.19012i
$306$	−2.19615	+	8.19615i	−0.125546	+	0.468543i
$307$	−18.0000	+	18.0000i	−1.02731	+	1.02731i	−0.0276979	+	0.999616i	$0.508818\pi$
$307$	−18.0000	+	18.0000i	−1.02731	+	1.02731i	−0.999616	+	0.0276979i	$0.991182\pi$
$308$		0			0
$309$		0			0
$310$	−16.0000	−	27.7128i	−0.908739	−	1.57398i
$311$	17.3205	+	10.0000i	0.982156	+	0.567048i	0.902920	−	0.429808i	$-0.141419\pi$
$311$	17.3205	+	10.0000i	0.982156	+	0.567048i	0.0792356	+	0.996856i	$0.474752\pi$
$312$		0			0
$313$	−12.1244	+	7.00000i	−0.685309	+	0.395663i	−0.801852	−	0.597522i	$-0.796152\pi$
$313$	−12.1244	+	7.00000i	−0.685309	+	0.395663i	0.116543	+	0.993186i	$0.462819\pi$
$314$			24.0000i			1.35440i
$315$		0			0
$316$		−	20.0000i		−	1.12509i
$317$	9.56218	+	2.56218i	0.537065	+	0.143906i	0.517149	−	0.855895i	$-0.326993\pi$
$317$	9.56218	+	2.56218i	0.537065	+	0.143906i	0.0199164	+	0.999802i	$0.493660\pi$
$318$		0			0
$319$	−7.00000	+	12.1244i	−0.391925	+	0.678834i
$320$	−21.8564	−	5.85641i	−1.22181	−	0.327383i
$321$		0			0
$322$		0			0
$323$	−4.00000	+	4.00000i	−0.222566	+	0.222566i
$324$	15.5885	−	9.00000i	0.866025	−	0.500000i
$325$		0			0
$326$	−25.9808	−	15.0000i	−1.43894	−	0.830773i
$327$		0			0
$328$	20.0000	−	20.0000i	1.10432	−	1.10432i
$329$		0			0
$330$		0			0
$331$	7.68653	−	28.6865i	0.422490	−	1.57675i	−0.346854	−	0.937919i	$-0.612750\pi$
$331$	7.68653	−	28.6865i	0.422490	−	1.57675i	0.769344	−	0.638835i	$-0.220583\pi$
$332$	27.3205	−	7.32051i	1.49941	−	0.401765i
$333$	5.49038	+	20.4904i	0.300871	+	1.12287i
$334$	4.39230	−	16.3923i	0.240336	−	0.896947i
$335$	−12.0000			−0.655630
$336$		0			0
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$	−4.75833	+	17.7583i	−0.258819	+	0.965926i
$339$		0			0
$340$	−2.92820	−	10.9282i	−0.158804	−	0.592665i
$341$	2.92820	−	10.9282i	0.158571	−	0.591795i
$342$	12.0000			0.648886
$343$		0			0
$344$		−	4.00000i		−	0.215666i
$345$		0			0
$346$		0			0
$347$	−20.4904	+	5.49038i	−1.09998	+	0.294739i	−0.762757	−	0.646685i	$-0.776155\pi$
$347$	−20.4904	+	5.49038i	−1.09998	+	0.294739i	−0.337224	+	0.941424i	$0.609488\pi$
$348$		0			0
$349$	−12.0000	+	12.0000i	−0.642345	+	0.642345i	−0.951131	−	0.308786i	$-0.900077\pi$
$349$	−12.0000	+	12.0000i	−0.642345	+	0.642345i	0.308786	+	0.951131i	$0.400077\pi$
$350$		0			0
$351$		0			0
$352$	−4.00000	−	6.92820i	−0.213201	−	0.369274i
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	−0.934751	−	0.355303i	$-0.884378\pi$
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	0.775077	+	0.631867i	$0.217711\pi$
$354$		0			0
$355$		0			0
$356$	28.0000			1.48400
$357$		0			0
$358$		−	6.00000i		−	0.317110i
$359$	5.19615	−	3.00000i	0.274242	−	0.158334i	−0.356572	−	0.934268i	$-0.616054\pi$
$359$	5.19615	−	3.00000i	0.274242	−	0.158334i	0.630814	+	0.775934i	$0.282721\pi$
$360$	−12.0000	+	20.7846i	−0.632456	+	1.09545i
$361$	−9.52628	−	5.50000i	−0.501383	−	0.289474i
$362$	4.00000	+	6.92820i	0.210235	+	0.364138i
$363$		0			0
$364$		0			0
$365$	12.0000	−	12.0000i	0.628109	−	0.628109i
$366$		0			0
$367$	−16.0000	+	27.7128i	−0.835193	+	1.44660i	0.0586798	+	0.998277i	$0.481311\pi$
$367$	−16.0000	+	27.7128i	−0.835193	+	1.44660i	−0.893873	+	0.448320i	$0.852022\pi$
$368$	−20.7846	−	12.0000i	−1.08347	−	0.625543i
$369$	−15.0000	−	25.9808i	−0.780869	−	1.35250i
$370$	−20.0000	−	20.0000i	−1.03975	−	1.03975i
$371$		0			0
$372$		0			0
$373$	−4.02628	+	15.0263i	−0.208473	+	0.778031i	0.779890	+	0.625917i	$0.215275\pi$
$373$	−4.02628	+	15.0263i	−0.208473	+	0.778031i	−0.988363	+	0.152115i	$0.951392\pi$
$374$	2.00000	−	3.46410i	0.103418	−	0.179124i
$375$		0			0
$376$	8.78461	−	32.7846i	0.453032	−	1.69074i
$377$		0			0
$378$		0			0
$379$	13.0000	+	13.0000i	0.667765	+	0.667765i	0.957198	−	0.289433i	$-0.0934668\pi$
$379$	13.0000	+	13.0000i	0.667765	+	0.667765i	−0.289433	+	0.957198i	$0.593467\pi$
$380$	−13.8564	+	8.00000i	−0.710819	+	0.410391i
$381$		0			0
$382$	6.58846	+	24.5885i	0.337095	+	1.25805i
$383$	2.00000	+	3.46410i	0.102195	+	0.177007i	0.912589	−	0.408879i	$-0.134080\pi$
$383$	2.00000	+	3.46410i	0.102195	+	0.177007i	−0.810394	+	0.585886i	$0.800747\pi$
$384$		0			0
$385$		0			0
$386$	16.0000	−	16.0000i	0.814379	−	0.814379i
$387$	−4.09808	−	1.09808i	−0.208317	−	0.0558184i
$388$	3.46410	+	2.00000i	0.175863	+	0.101535i
$389$	−4.09808	+	1.09808i	−0.207781	+	0.0556747i	−0.361208	−	0.932485i	$-0.617636\pi$
$389$	−4.09808	+	1.09808i	−0.207781	+	0.0556747i	0.153427	+	0.988160i	$0.450969\pi$
$390$		0			0
$391$		−	12.0000i		−	0.606866i
$392$		0			0
$393$		0			0
$394$	5.00000	+	8.66025i	0.251896	+	0.436297i
$395$	27.3205	−	7.32051i	1.37464	−	0.368335i
$396$	−8.19615	+	2.19615i	−0.411872	+	0.110361i
$397$	−16.3923	−	4.39230i	−0.822706	−	0.220443i	−0.177177	−	0.984179i	$-0.556696\pi$
$397$	−16.3923	−	4.39230i	−0.822706	−	0.220443i	−0.645529	+	0.763736i	$0.723363\pi$
$398$	−24.0000	−	24.0000i	−1.20301	−	1.20301i
$399$		0			0
$400$		−	12.0000i		−	0.600000i
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	0.676425	−	0.736512i	$-0.263528\pi$
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	−0.976050	+	0.217545i	$0.930195\pi$
$402$		0			0
$403$		0			0
$404$	16.3923	+	4.39230i	0.815548	+	0.218525i
$405$	18.0000	+	18.0000i	0.894427	+	0.894427i
$406$		0			0
$407$		−	10.0000i		−	0.495682i
$408$		0			0
$409$	−12.1244	−	7.00000i	−0.599511	−	0.346128i	0.169338	−	0.985558i	$-0.445837\pi$
$409$	−12.1244	−	7.00000i	−0.599511	−	0.346128i	−0.768849	+	0.639430i	$0.779170\pi$
$410$	34.6410	+	20.0000i	1.71080	+	0.987730i
$411$		0			0
$412$	−8.00000			−0.394132
$413$		0			0
$414$	−18.0000	+	18.0000i	−0.884652	+	0.884652i
$415$	20.0000	+	34.6410i	0.981761	+	1.70046i
$416$		0			0
$417$		0			0
$418$	−5.46410	−	1.46410i	−0.267258	−	0.0716116i
$419$	−12.0000	+	12.0000i	−0.586238	+	0.586238i	−0.936611	−	0.350372i	$-0.886055\pi$
$419$	−12.0000	+	12.0000i	−0.586238	+	0.586238i	0.350372	+	0.936611i	$0.386055\pi$
$420$		0			0
$421$	−9.00000	−	9.00000i	−0.438633	−	0.438633i	0.452919	−	0.891552i	$-0.350383\pi$
$421$	−9.00000	−	9.00000i	−0.438633	−	0.438633i	−0.891552	+	0.452919i	$0.850383\pi$
$422$	−15.5885	+	9.00000i	−0.758834	+	0.438113i
$423$	−31.1769	−	18.0000i	−1.51587	−	0.875190i
$424$	3.46410	+	2.00000i	0.168232	+	0.0971286i
$425$	5.19615	−	3.00000i	0.252050	−	0.145521i
$426$		0			0
$427$		0			0
$428$	10.0000	−	10.0000i	0.483368	−	0.483368i
$429$		0			0
$430$	5.46410	−	1.46410i	0.263502	−	0.0706052i
$431$	4.00000	−	6.92820i	0.192673	−	0.333720i	−0.753462	−	0.657491i	$-0.771618\pi$
$431$	4.00000	−	6.92820i	0.192673	−	0.333720i	0.946135	+	0.323772i	$0.104951\pi$
$432$		0			0
$433$	6.00000			0.288342			0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000			0.288342			0.144171	+	0.989553i	$0.453949\pi$
$434$		0			0
$435$		0			0
$436$	2.19615	−	8.19615i	0.105177	−	0.392525i
$437$	−16.3923	+	4.39230i	−0.784150	+	0.210112i
$438$		0			0
$439$	−13.8564	+	8.00000i	−0.661330	+	0.381819i	−0.792784	−	0.609503i	$-0.791369\pi$
$439$	−13.8564	+	8.00000i	−0.661330	+	0.381819i	0.131453	+	0.991322i	$0.458036\pi$
$440$	8.00000	−	8.00000i	0.381385	−	0.381385i
$441$		0			0
$442$		0			0
$443$	−0.366025	+	1.36603i	−0.0173904	+	0.0649018i	−0.974076	−	0.226222i	$-0.927363\pi$
$443$	−0.366025	+	1.36603i	−0.0173904	+	0.0649018i	0.956685	+	0.291124i	$0.0940292\pi$
$444$		0			0
$445$	10.2487	+	38.2487i	0.485836	+	1.81316i
$446$	−5.46410	−	1.46410i	−0.258733	−	0.0693272i
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	−12.2942	−	3.29423i	−0.579555	−	0.155291i
$451$	3.66025	+	13.6603i	0.172355	+	0.643236i
$452$	6.92820	+	4.00000i	0.325875	+	0.188144i
$453$		0			0
$454$			4.00000i			0.187729i
$455$		0			0
$456$		0			0
$457$	19.0526	−	11.0000i	0.891241	−	0.514558i	0.0168929	−	0.999857i	$-0.494623\pi$
$457$	19.0526	−	11.0000i	0.891241	−	0.514558i	0.874348	+	0.485299i	$0.161289\pi$
$458$	−8.00000	+	13.8564i	−0.373815	+	0.647467i
$459$		0			0
$460$	8.78461	−	32.7846i	0.409585	−	1.52859i
$461$	−16.0000	+	16.0000i	−0.745194	+	0.745194i	−0.973572	−	0.228378i	$-0.926658\pi$
$461$	−16.0000	+	16.0000i	−0.745194	+	0.745194i	0.228378	+	0.973572i	$0.426658\pi$
$462$		0			0
$463$	14.0000			0.650635			0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000			0.650635			0.325318	+	0.945605i	$0.394529\pi$
$464$	38.2487	+	10.2487i	1.77565	+	0.475784i
$465$		0			0
$466$	−21.8564	+	5.85641i	−1.01248	+	0.271293i
$467$	24.5885	+	6.58846i	1.13782	+	0.304877i	0.778073	−	0.628173i	$-0.216197\pi$
$467$	24.5885	+	6.58846i	1.13782	+	0.304877i	0.359745	+	0.933051i	$0.382864\pi$
$468$		0			0
$469$		0			0
$470$	48.0000			2.21407
$471$		0			0
$472$	27.7128	+	16.0000i	1.27559	+	0.736460i
$473$	1.73205	+	1.00000i	0.0796398	+	0.0459800i
$474$		0			0
$475$	−6.00000	−	6.00000i	−0.275299	−	0.275299i
$476$		0			0
$477$	3.00000	−	3.00000i	0.137361	−	0.137361i
$478$		0			0
$479$	−10.0000	+	17.3205i	−0.456912	+	0.791394i	−0.998796	−	0.0490589i	$-0.984378\pi$
$479$	−10.0000	+	17.3205i	−0.456912	+	0.791394i	0.541884	+	0.840453i	$0.317711\pi$
$480$		0			0
$481$		0			0
$482$	22.0000	−	22.0000i	1.00207	−	1.00207i
$483$		0			0
$484$	−18.0000			−0.818182
$485$	−1.46410	+	5.46410i	−0.0664814	+	0.248112i
$486$		0			0
$487$	−19.0526	−	11.0000i	−0.863354	−	0.498458i	0.00178012	−	0.999998i	$-0.499433\pi$
$487$	−19.0526	−	11.0000i	−0.863354	−	0.498458i	−0.865134	+	0.501541i	$0.832767\pi$
$488$	12.0000	+	20.7846i	0.543214	+	0.940875i
$489$		0			0
$490$		0			0
$491$	−19.0000	−	19.0000i	−0.857458	−	0.857458i	0.133580	−	0.991038i	$-0.457353\pi$
$491$	−19.0000	−	19.0000i	−0.857458	−	0.857458i	−0.991038	+	0.133580i	$0.957353\pi$
$492$		0			0
$493$	5.12436	+	19.1244i	0.230789	+	0.861318i
$494$		0			0
$495$	−6.00000	−	10.3923i	−0.269680	−	0.467099i
$496$	−32.0000			−1.43684
$497$		0			0
$498$		0			0
$499$	31.4186	+	8.41858i	1.40649	+	0.376868i	0.880671	−	0.473729i	$-0.157092\pi$
$499$	31.4186	+	8.41858i	1.40649	+	0.376868i	0.525818	+	0.850597i	$0.323759\pi$
$500$	−10.9282	+	2.92820i	−0.488724	+	0.130953i
$501$		0			0
$502$	14.0000	+	24.2487i	0.624851	+	1.08227i
$503$			16.0000i			0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$			16.0000i			0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$		0			0
$505$			24.0000i			1.06799i
$506$	10.3923	−	6.00000i	0.461994	−	0.266733i
$507$		0			0
$508$	13.8564	+	8.00000i	0.614779	+	0.354943i
$509$	−10.9282	−	2.92820i	−0.484384	−	0.129790i	0.00835918	−	0.999965i	$-0.497339\pi$
$509$	−10.9282	−	2.92820i	−0.484384	−	0.129790i	−0.492743	+	0.870175i	$0.664006\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$		0			0
$514$	−0.732051	−	2.73205i	−0.0322894	−	0.120506i
$515$	−2.92820	−	10.9282i	−0.129032	−	0.481554i
$516$		0			0
$517$	12.0000	+	12.0000i	0.527759	+	0.527759i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.66025	−	5.00000i	−0.379413	−	0.219054i	0.298150	−	0.954519i	$-0.403630\pi$
$521$	−8.66025	−	5.00000i	−0.379413	−	0.219054i	−0.677563	+	0.735465i	$0.736964\pi$
$522$	21.0000	−	36.3731i	0.919145	−	1.59201i
$523$	−8.78461	+	32.7846i	−0.384124	+	1.43357i	0.455419	+	0.890277i	$0.349490\pi$
$523$	−8.78461	+	32.7846i	−0.384124	+	1.43357i	−0.839543	+	0.543293i	$0.817177\pi$
$524$	−28.0000	−	28.0000i	−1.22319	−	1.22319i
$525$		0			0
$526$	16.0000	+	16.0000i	0.697633	+	0.697633i
$527$	−8.00000	−	13.8564i	−0.348485	−	0.603595i
$528$		0			0
$529$	6.50000	−	11.2583i	0.282609	−	0.489493i
$530$	−1.46410	+	5.46410i	−0.0635965	+	0.237345i
$531$	24.0000	−	24.0000i	1.04151	−	1.04151i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	17.3205	+	10.0000i	0.748831	+	0.432338i
$536$	−6.00000	+	10.3923i	−0.259161	+	0.448879i
$537$		0			0
$538$			16.0000i			0.689809i
$539$		0			0
$540$		0			0
$541$	−15.0263	−	4.02628i	−0.646030	−	0.173103i	−0.0790969	−	0.996867i	$-0.525204\pi$
$541$	−15.0263	−	4.02628i	−0.646030	−	0.173103i	−0.566933	+	0.823764i	$0.691870\pi$
$542$	−2.92820	−	10.9282i	−0.125777	−	0.469407i
$543$		0			0
$544$	−10.9282	−	2.92820i	−0.468543	−	0.125546i
$545$	12.0000			0.514024
$546$		0			0
$547$	23.0000	−	23.0000i	0.983409	−	0.983409i	−0.0164556	−	0.999865i	$-0.505238\pi$
$547$	23.0000	−	23.0000i	0.983409	−	0.983409i	0.999865	+	0.0164556i	$0.00523822\pi$
$548$	−12.0000	−	20.7846i	−0.512615	−	0.887875i
$549$	24.5885	−	6.58846i	1.04941	−	0.281189i
$550$	5.19615	+	3.00000i	0.221565	+	0.127920i
$551$	24.2487	−	14.0000i	1.03303	−	0.596420i
$552$		0			0
$553$		0			0
$554$	30.0000			1.27458
$555$		0			0
$556$	1.46410	+	5.46410i	0.0620917	+	0.231730i
$557$	−6.22243	−	23.2224i	−0.263653	−	0.983966i	−0.963070	−	0.269252i	$-0.913224\pi$
$557$	−6.22243	−	23.2224i	−0.263653	−	0.983966i	0.699417	−	0.714714i	$-0.253443\pi$
$558$	−8.78461	+	32.7846i	−0.371882	+	1.38788i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.32051	−	27.3205i	−0.308523	−	1.15142i	−0.929870	−	0.367887i	$-0.880081\pi$
$563$	−7.32051	−	27.3205i	−0.308523	−	1.15142i	0.621348	−	0.783535i	$-0.286585\pi$
$564$		0			0
$565$	−2.92820	+	10.9282i	−0.123190	+	0.459753i
$566$	8.00000			0.336265
$567$		0			0
$568$		0			0
$569$	−12.1244	+	7.00000i	−0.508279	+	0.293455i	−0.732126	−	0.681169i	$-0.761472\pi$
$569$	−12.1244	+	7.00000i	−0.508279	+	0.293455i	0.223847	+	0.974624i	$0.428139\pi$
$570$		0			0
$571$	12.2942	−	3.29423i	0.514497	−	0.137859i	0.00777727	−	0.999970i	$-0.497524\pi$
$571$	12.2942	−	3.29423i	0.514497	−	0.137859i	0.506720	+	0.862111i	$0.330858\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.0000			0.750652
$576$	12.0000	+	20.7846i	0.500000	+	0.866025i
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	−0.886090	−	0.463513i	$-0.846589\pi$
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	0.844459	+	0.535620i	$0.179922\pi$
$578$	4.75833	+	17.7583i	0.197920	+	0.738649i
$579$		0			0
$580$			56.0000i			2.32527i
$581$		0			0
$582$		0			0
$583$	−1.73205	+	1.00000i	−0.0717342	+	0.0414158i
$584$	−4.39230	−	16.3923i	−0.181755	−	0.678318i
$585$		0			0
$586$	−14.0000	−	24.2487i	−0.578335	−	1.00171i
$587$	−10.0000	−	10.0000i	−0.412744	−	0.412744i	0.469949	−	0.882693i	$-0.344272\pi$
$587$	−10.0000	−	10.0000i	−0.412744	−	0.412744i	−0.882693	+	0.469949i	$0.844272\pi$
$588$		0			0
$589$	−16.0000	+	16.0000i	−0.659269	+	0.659269i
$590$	−11.7128	+	43.7128i	−0.482209	+	1.79963i
$591$		0			0
$592$	−27.3205	+	7.32051i	−1.12287	+	0.300871i
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	0.797831	−	0.602881i	$-0.205981\pi$
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	−0.921026	+	0.389501i	$0.872647\pi$
$594$		0			0
$595$		0			0
$596$	6.00000	−	6.00000i	0.245770	−	0.245770i
$597$		0			0
$598$		0			0
$599$	20.7846	+	12.0000i	0.849236	+	0.490307i	0.860393	−	0.509631i	$-0.170218\pi$
$599$	20.7846	+	12.0000i	0.849236	+	0.490307i	−0.0111569	+	0.999938i	$0.503551\pi$
$600$		0			0
$601$		−	10.0000i		−	0.407909i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$		−	10.0000i		−	0.407909i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$		0			0
$603$	9.00000	+	9.00000i	0.366508	+	0.366508i
$604$	−10.0000	−	17.3205i	−0.406894	−	0.704761i
$605$	−6.58846	−	24.5885i	−0.267859	−	0.999663i
$606$		0			0
$607$	14.0000	+	24.2487i	0.568242	+	0.984225i	0.996740	+	0.0806818i	$0.0257098\pi$
$607$	14.0000	+	24.2487i	0.568242	+	0.984225i	−0.428497	+	0.903543i	$0.640957\pi$
$608$			16.0000i			0.648886i
$609$		0			0
$610$	−24.0000	+	24.0000i	−0.971732	+	0.971732i
$611$		0			0
$612$	−6.00000	+	10.3923i	−0.242536	+	0.420084i
$613$	−12.2942	+	3.29423i	−0.496559	+	0.133053i	−0.498403	−	0.866946i	$-0.666080\pi$
$613$	−12.2942	+	3.29423i	−0.496559	+	0.133053i	0.00184345	+	0.999998i	$0.499413\pi$
$614$	−31.1769	+	18.0000i	−1.25820	+	0.726421i
$615$		0			0
$616$		0			0
$617$			42.0000i			1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$			42.0000i			1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$		0			0
$619$	−16.3923	+	4.39230i	−0.658862	+	0.176542i	−0.572733	−	0.819742i	$-0.694117\pi$
$619$	−16.3923	+	4.39230i	−0.658862	+	0.176542i	−0.0861298	+	0.996284i	$0.527450\pi$
$620$	−11.7128	−	43.7128i	−0.470398	−	1.75555i
$621$		0			0
$622$	20.0000	+	20.0000i	0.801927	+	0.801927i
$623$		0			0
$624$		0			0
$625$	−15.5000	−	26.8468i	−0.620000	−	1.07387i
$626$	−19.1244	+	5.12436i	−0.764363	+	0.204810i
$627$		0			0
$628$	−8.78461	+	32.7846i	−0.350544	+	1.30825i
$629$	−10.0000	−	10.0000i	−0.398726	−	0.398726i
$630$		0			0
$631$		−	40.0000i		−	1.59237i	−0.605050	−	0.796187i	$-0.706847\pi$
$631$		−	40.0000i		−	1.59237i	0.605050	−	0.796187i	$-0.293153\pi$
$632$	7.32051	−	27.3205i	0.291194	−	1.08675i
$633$		0			0
$634$	12.1244	+	7.00000i	0.481520	+	0.278006i
$635$	−5.85641	+	21.8564i	−0.232404	+	0.867345i
$636$		0			0
$637$		0			0
$638$	−14.0000	+	14.0000i	−0.554265	+	0.554265i
$639$		0			0
$640$	−27.7128	−	16.0000i	−1.09545	−	0.632456i
$641$	−6.00000	+	10.3923i	−0.236986	+	0.410471i	−0.959848	−	0.280521i	$-0.909493\pi$
$641$	−6.00000	+	10.3923i	−0.236986	+	0.410471i	0.722862	+	0.690992i	$0.242826\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$	−6.92820	+	4.00000i	−0.272587	+	0.157378i
$647$	10.3923	+	6.00000i	0.408564	+	0.235884i	0.690172	−	0.723645i	$-0.257535\pi$
$647$	10.3923	+	6.00000i	0.408564	+	0.235884i	−0.281609	+	0.959529i	$0.590868\pi$
$648$	24.5885	−	6.58846i	0.965926	−	0.258819i
$649$	−13.8564	+	8.00000i	−0.543912	+	0.314027i
$650$		0			0
$651$		0			0
$652$	−30.0000	−	30.0000i	−1.17489	−	1.17489i
$653$	−34.1506	−	9.15064i	−1.33642	−	0.358092i	−0.481314	−	0.876548i	$-0.659840\pi$
$653$	−34.1506	−	9.15064i	−1.33642	−	0.358092i	−0.855104	+	0.518456i	$0.826507\pi$
$654$		0			0
$655$	28.0000	−	48.4974i	1.09405	−	1.89495i
$656$	34.6410	−	20.0000i	1.35250	−	0.780869i
$657$	−18.0000			−0.702247
$658$		0			0
$659$	−3.00000	+	3.00000i	−0.116863	+	0.116863i	−0.763120	−	0.646257i	$-0.776334\pi$
$659$	−3.00000	+	3.00000i	−0.116863	+	0.116863i	0.646257	+	0.763120i	$0.276334\pi$
$660$		0			0
$661$	35.5167	−	9.51666i	1.38144	−	0.370155i	0.509795	−	0.860296i	$-0.329721\pi$
$661$	35.5167	−	9.51666i	1.38144	−	0.370155i	0.871643	+	0.490141i	$0.163055\pi$
$662$	21.0000	−	36.3731i	0.816188	−	1.41368i
$663$		0			0
$664$	40.0000			1.55230
$665$		0			0
$666$			30.0000i			1.16248i
$667$	−15.3731	+	57.3731i	−0.595248	+	2.22149i
$668$	12.0000	−	20.7846i	0.464294	−	0.804181i
$669$		0			0
$670$	−16.3923	−	4.39230i	−0.633290	−	0.169690i
$671$	−12.0000			−0.463255
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$	10.9282	+	2.92820i	0.420939	+	0.112790i
$675$		0			0
$676$	−13.0000	+	22.5167i	−0.500000	+	0.866025i
$677$	7.32051	−	27.3205i	0.281350	−	1.05001i	−0.670115	−	0.742257i	$-0.733755\pi$
$677$	7.32051	−	27.3205i	0.281350	−	1.05001i	0.951465	−	0.307756i	$-0.0995779\pi$
$678$		0			0
$679$		0			0
$680$		−	16.0000i		−	0.613572i
$681$		0			0
$682$	8.00000	−	13.8564i	0.306336	−	0.530589i
$683$	15.0263	−	4.02628i	0.574965	−	0.154061i	0.0403921	−	0.999184i	$-0.487139\pi$
$683$	15.0263	−	4.02628i	0.574965	−	0.154061i	0.534573	+	0.845123i	$0.320473\pi$
$684$	16.3923	+	4.39230i	0.626775	+	0.167944i
$685$	24.0000	−	24.0000i	0.916993	−	0.916993i
$686$		0			0
$687$		0			0
$688$	1.46410	−	5.46410i	0.0558184	−	0.208317i
$689$		0			0
$690$		0			0
$691$	35.5167	+	9.51666i	1.35112	+	0.362031i	0.860547	−	0.509371i	$-0.170122\pi$
$691$	35.5167	+	9.51666i	1.35112	+	0.362031i	0.490570	+	0.871402i	$0.336788\pi$
$692$		0			0
$693$		0			0
$694$	−30.0000			−1.13878
$695$	−6.92820	+	4.00000i	−0.262802	+	0.151729i
$696$		0			0
$697$	17.3205	+	10.0000i	0.656061	+	0.378777i
$698$	−20.7846	+	12.0000i	−0.786709	+	0.454207i
$699$		0			0
$700$		0			0
$701$	−19.0000	+	19.0000i	−0.717620	+	0.717620i	−0.968117	−	0.250497i	$-0.919406\pi$
$701$	−19.0000	+	19.0000i	−0.717620	+	0.717620i	0.250497	+	0.968117i	$0.419406\pi$
$702$		0			0
$703$	−10.0000	+	17.3205i	−0.377157	+	0.653255i
$704$	−2.92820	−	10.9282i	−0.110361	−	0.411872i
$705$		0			0
$706$	−6.00000	+	6.00000i	−0.225813	+	0.225813i
$707$		0			0
$708$		0			0
$709$	−2.56218	+	9.56218i	−0.0962246	+	0.359115i	−0.997202	−	0.0747503i	$-0.976184\pi$
$709$	−2.56218	+	9.56218i	−0.0962246	+	0.359115i	0.900978	+	0.433865i	$0.142851\pi$
$710$		0			0
$711$	−25.9808	−	15.0000i	−0.974355	−	0.562544i
$712$	38.2487	+	10.2487i	1.43343	+	0.384087i
$713$		−	48.0000i		−	1.79761i
$714$		0			0
$715$		0			0
$716$	2.19615	−	8.19615i	0.0820741	−	0.306305i
$717$		0			0
$718$	8.19615	−	2.19615i	0.305878	−	0.0819597i
$719$	10.0000	+	17.3205i	0.372937	+	0.645946i	0.990016	−	0.140955i	$-0.0450174\pi$
$719$	10.0000	+	17.3205i	0.372937	+	0.645946i	−0.617079	+	0.786901i	$0.711684\pi$
$720$	−24.0000	+	24.0000i	−0.894427	+	0.894427i
$721$		0			0
$722$	−11.0000	−	11.0000i	−0.409378	−	0.409378i
$723$		0			0
$724$	2.92820	+	10.9282i	0.108826	+	0.406143i
$725$	−28.6865	+	7.68653i	−1.06539	+	0.285471i
$726$		0			0
$727$		−	32.0000i		−	1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$		−	32.0000i		−	1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$	20.7846	−	12.0000i	0.769273	−	0.444140i
$731$	2.73205	−	0.732051i	0.101049	−	0.0270759i
$732$		0			0
$733$	−13.6603	−	3.66025i	−0.504553	−	0.135195i	−0.00244074	−	0.999997i	$-0.500777\pi$
$733$	−13.6603	−	3.66025i	−0.504553	−	0.135195i	−0.502112	+	0.864802i	$0.667444\pi$
$734$	−32.0000	+	32.0000i	−1.18114	+	1.18114i
$735$		0			0
$736$	−24.0000	−	24.0000i	−0.884652	−	0.884652i
$737$	−3.00000	−	5.19615i	−0.110506	−	0.191403i
$738$	−10.9808	−	40.9808i	−0.404207	−	1.50852i
$739$	2.56218	+	9.56218i	0.0942512	+	0.351750i	0.996905	−	0.0786174i	$-0.0250505\pi$
$739$	2.56218	+	9.56218i	0.0942512	+	0.351750i	−0.902654	+	0.430368i	$0.858384\pi$
$740$	−20.0000	−	34.6410i	−0.735215	−	1.27343i
$741$		0			0
$742$		0			0
$743$		−	6.00000i		−	0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$		−	6.00000i		−	0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$		0			0
$745$	10.3923	+	6.00000i	0.380745	+	0.219823i
$746$	−11.0000	+	19.0526i	−0.402739	+	0.697564i
$747$	10.9808	−	40.9808i	0.401765	−	1.49941i
$748$	4.00000	−	4.00000i	0.146254	−	0.146254i
$749$		0			0
$750$		0			0
$751$	−16.0000	−	27.7128i	−0.583848	−	1.01125i	−0.995018	−	0.0996961i	$-0.968213\pi$
$751$	−16.0000	−	27.7128i	−0.583848	−	1.01125i	0.411170	−	0.911559i	$-0.365120\pi$
$752$	24.0000	−	41.5692i	0.875190	−	1.51587i
$753$		0			0
$754$		0			0
$755$	20.0000	−	20.0000i	0.727875	−	0.727875i
$756$		0			0
$757$	−15.0000	−	15.0000i	−0.545184	−	0.545184i	0.379860	−	0.925044i	$-0.375972\pi$
$757$	−15.0000	−	15.0000i	−0.545184	−	0.545184i	−0.925044	+	0.379860i	$0.875972\pi$
$758$	13.0000	+	22.5167i	0.472181	+	0.817842i
$759$		0			0
$760$	−21.8564	+	5.85641i	−0.792815	+	0.212434i
$761$	25.9808	−	15.0000i	0.941802	−	0.543750i	0.0512772	−	0.998684i	$-0.483671\pi$
$761$	25.9808	−	15.0000i	0.941802	−	0.543750i	0.890525	+	0.454935i	$0.150337\pi$
$762$		0			0
$763$		0			0
$764$			36.0000i			1.30243i
$765$	−16.3923	−	4.39230i	−0.592665	−	0.158804i
$766$	1.46410	+	5.46410i	0.0529001	+	0.197426i
$767$		0			0
$768$		0			0
$769$	10.0000			0.360609			0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000			0.360609			0.180305	+	0.983611i	$0.442292\pi$
$770$		0			0
$771$		0			0
$772$	27.7128	−	16.0000i	0.997406	−	0.575853i
$773$	−8.19615	+	2.19615i	−0.294795	+	0.0789901i	−0.403185	−	0.915118i	$-0.632097\pi$
$773$	−8.19615	+	2.19615i	−0.294795	+	0.0789901i	0.108390	+	0.994108i	$0.465430\pi$
$774$	−5.19615	−	3.00000i	−0.186772	−	0.107833i
$775$	20.7846	−	12.0000i	0.746605	−	0.431053i
$776$	4.00000	+	4.00000i	0.143592	+	0.143592i
$777$		0			0
$778$	−6.00000			−0.215110
$779$	7.32051	−	27.3205i	0.262284	−	0.978859i
$780$		0			0
$781$		0			0
$782$	4.39230	−	16.3923i	0.157069	−	0.586188i
$783$		0			0
$784$		0			0
$785$	−48.0000			−1.71319
$786$		0			0
$787$	−6.58846	−	24.5885i	−0.234853	−	0.876484i	−0.978215	−	0.207594i	$-0.933437\pi$
$787$	−6.58846	−	24.5885i	−0.234853	−	0.876484i	0.743362	−	0.668889i	$-0.233230\pi$
$788$	3.66025	+	13.6603i	0.130391	+	0.486626i
$789$		0			0
$790$	40.0000			1.42314
$791$		0			0
$792$	−12.0000			−0.426401
$793$		0			0
$794$	−20.7846	−	12.0000i	−0.737618	−	0.425864i
$795$		0			0
$796$	−24.0000	−	41.5692i	−0.850657	−	1.47338i
$797$	32.0000	−	32.0000i	1.13350	−	1.13350i	0.143907	−	0.989591i	$-0.454033\pi$
$797$	32.0000	−	32.0000i	1.13350	−	1.13350i	0.989591	−	0.143907i	$-0.0459666\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$	4.39230	−	16.3923i	0.155291	−	0.579555i
$801$	21.0000	−	36.3731i	0.741999	−	1.28518i
$802$	−4.39230	−	16.3923i	−0.155098	−	0.578832i
$803$	8.19615	+	2.19615i	0.289236	+	0.0775005i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	20.7846	+	12.0000i	0.731200	+	0.422159i
$809$	−5.19615	−	3.00000i	−0.182687	−	0.105474i	0.405868	−	0.913932i	$-0.366969\pi$
$809$	−5.19615	−	3.00000i	−0.182687	−	0.105474i	−0.588555	+	0.808458i	$0.700303\pi$
$810$	18.0000	+	31.1769i	0.632456	+	1.09545i
$811$	24.0000	+	24.0000i	0.842754	+	0.842754i	0.989216	−	0.146462i	$-0.0467887\pi$
$811$	24.0000	+	24.0000i	0.842754	+	0.842754i	−0.146462	+	0.989216i	$0.546789\pi$
$812$		0			0
$813$		0			0
$814$	3.66025	−	13.6603i	0.128292	−	0.478792i
$815$	30.0000	−	51.9615i	1.05085	−	1.82013i
$816$		0			0
$817$	−2.00000	−	3.46410i	−0.0699711	−	0.121194i
$818$	−14.0000	−	14.0000i	−0.489499	−	0.489499i
$819$		0			0
$820$	40.0000	+	40.0000i	1.39686	+	1.39686i
$821$	0.366025	−	1.36603i	0.0127744	−	0.0476746i	−0.959244	−	0.282578i	$-0.908811\pi$
$821$	0.366025	−	1.36603i	0.0127744	−	0.0476746i	0.972019	+	0.234903i	$0.0754772\pi$
$822$		0			0
$823$	−20.7846	−	12.0000i	−0.724506	−	0.418294i	0.0919029	−	0.995768i	$-0.470705\pi$
$823$	−20.7846	−	12.0000i	−0.724506	−	0.418294i	−0.816409	+	0.577474i	$0.804038\pi$
$824$	−10.9282	−	2.92820i	−0.380702	−	0.102009i
$825$		0			0
$826$		0			0
$827$	−5.00000	−	5.00000i	−0.173867	−	0.173867i	0.614809	−	0.788676i	$-0.289233\pi$
$827$	−5.00000	−	5.00000i	−0.173867	−	0.173867i	−0.788676	+	0.614809i	$0.789233\pi$
$828$	−31.1769	+	18.0000i	−1.08347	+	0.625543i
$829$	−0.732051	−	2.73205i	−0.0254252	−	0.0948880i	0.952047	−	0.305951i	$-0.0989743\pi$
$829$	−0.732051	−	2.73205i	−0.0254252	−	0.0948880i	−0.977473	+	0.211063i	$0.932308\pi$
$830$	14.6410	+	54.6410i	0.508197	+	1.89662i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	32.7846	+	8.78461i	1.13456	+	0.304004i
$836$	−6.92820	−	4.00000i	−0.239617	−	0.138343i
$837$		0			0
$838$	−20.7846	+	12.0000i	−0.717992	+	0.414533i
$839$			4.00000i			0.138095i	0.997613	+	0.0690477i	$0.0219961\pi$
$839$			4.00000i			0.138095i	−0.997613	+	0.0690477i	$0.978004\pi$
$840$		0			0
$841$		−	69.0000i		−	2.37931i
$842$	−9.00000	−	15.5885i	−0.310160	−	0.537214i
$843$		0			0
$844$	−24.5885	+	6.58846i	−0.846370	+	0.226784i
$845$	−35.5167	−	9.51666i	−1.22181	−	0.327383i
$846$	−36.0000	−	36.0000i	−1.23771	−	1.23771i
$847$		0			0
$848$	4.00000	+	4.00000i	0.137361	+	0.137361i
$849$		0			0
$850$	8.19615	−	2.19615i	0.281126	−	0.0753274i
$851$	−10.9808	−	40.9808i	−0.376416	−	1.40480i
$852$		0			0
$853$	−24.0000	−	24.0000i	−0.821744	−	0.821744i	0.164614	−	0.986358i	$-0.447362\pi$
$853$	−24.0000	−	24.0000i	−0.821744	−	0.821744i	−0.986358	+	0.164614i	$0.947362\pi$
$854$		0			0
$855$			24.0000i			0.820783i
$856$	17.3205	−	10.0000i	0.592003	−	0.341793i
$857$	−15.5885	−	9.00000i	−0.532492	−	0.307434i	0.209539	−	0.977800i	$-0.432804\pi$
$857$	−15.5885	−	9.00000i	−0.532492	−	0.307434i	−0.742030	+	0.670366i	$0.766137\pi$
$858$		0			0
$859$	0.732051	−	2.73205i	0.0249773	−	0.0932164i	−0.952312	−	0.305126i	$-0.901302\pi$
$859$	0.732051	−	2.73205i	0.0249773	−	0.0932164i	0.977289	+	0.211909i	$0.0679682\pi$
$860$	8.00000			0.272798
$861$		0			0
$862$	8.00000	−	8.00000i	0.272481	−	0.272481i
$863$	−7.00000	−	12.1244i	−0.238283	−	0.412718i	0.721939	−	0.691957i	$-0.243251\pi$
$863$	−7.00000	−	12.1244i	−0.238283	−	0.412718i	−0.960222	+	0.279239i	$0.909918\pi$
$864$		0			0
$865$		0			0
$866$	8.19615	+	2.19615i	0.278517	+	0.0746283i
$867$		0			0
$868$		0			0
$869$	10.0000	+	10.0000i	0.339227	+	0.339227i
$870$		0			0
$871$		0			0
$872$	6.00000	−	10.3923i	0.203186	−	0.351928i
$873$	5.19615	−	3.00000i	0.175863	−	0.101535i
$874$	−24.0000			−0.811812
$875$		0			0
$876$		0			0
$877$	−4.09808	−	1.09808i	−0.138382	−	0.0370794i	0.188963	−	0.981984i	$-0.439487\pi$
$877$	−4.09808	−	1.09808i	−0.138382	−	0.0370794i	−0.327345	+	0.944905i	$0.606154\pi$
$878$	−21.8564	+	5.85641i	−0.737618	+	0.197644i
$879$		0			0
$880$	13.8564	−	8.00000i	0.467099	−	0.269680i
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	−25.0000	+	25.0000i	−0.841317	+	0.841317i	−0.989030	−	0.147713i	$-0.952809\pi$
$883$	−25.0000	+	25.0000i	−0.841317	+	0.841317i	0.147713	+	0.989030i	$0.452809\pi$
$884$		0			0
$885$		0			0
$886$	−1.00000	+	1.73205i	−0.0335957	+	0.0581894i
$887$	−10.3923	+	6.00000i	−0.348939	+	0.201460i	−0.664218	−	0.747539i	$-0.731235\pi$
$887$	−10.3923	+	6.00000i	−0.348939	+	0.201460i	0.315279	+	0.948999i	$0.397902\pi$
$888$		0			0
$889$		0			0
$890$			56.0000i			1.87712i
$891$	−3.29423	+	12.2942i	−0.110361	+	0.411872i
$892$	−6.92820	−	4.00000i	−0.231973	−	0.133930i
$893$	−8.78461	−	32.7846i	−0.293966	−	1.09710i
$894$		0			0
$895$	12.0000			0.401116
$896$		0			0
$897$		0			0
$898$	−40.9808	−	10.9808i	−1.36755	−	0.366433i
$899$	20.4974	+	76.4974i	0.683627	+	2.55133i
$900$	−15.5885	−	9.00000i	−0.519615	−	0.300000i
$901$	−0.732051	+	2.73205i	−0.0243881	+	0.0910178i
$902$			20.0000i			0.665927i
$903$		0			0
$904$	8.00000	+	8.00000i	0.266076	+	0.266076i
$905$	−13.8564	+	8.00000i	−0.460603	+	0.265929i
$906$		0			0
$907$	−20.4904	+	5.49038i	−0.680372	+	0.182305i	−0.582422	−	0.812886i	$-0.697895\pi$
$907$	−20.4904	+	5.49038i	−0.680372	+	0.182305i	−0.0979495	+	0.995191i	$0.531228\pi$
$908$	−1.46410	+	5.46410i	−0.0485879	+	0.181333i
$909$	18.0000	−	18.0000i	0.597022	−	0.597022i
$910$		0			0
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$		0			0
$913$	−10.0000	+	17.3205i	−0.330952	+	0.573225i
$914$	30.0526	−	8.05256i	0.994050	−	0.266355i
$915$		0			0
$916$	−16.0000	+	16.0000i	−0.528655	+	0.528655i
$917$		0			0
$918$		0			0
$919$	−20.7846	+	12.0000i	−0.685621	+	0.395843i	−0.801970	−	0.597365i	$-0.796214\pi$
$919$	−20.7846	+	12.0000i	−0.685621	+	0.395843i	0.116348	+	0.993208i	$0.462881\pi$
$920$	24.0000	−	41.5692i	0.791257	−	1.37050i
$921$		0			0
$922$	−27.7128	+	16.0000i	−0.912673	+	0.526932i
$923$		0			0
$924$		0			0
$925$	15.0000	−	15.0000i	0.493197	−	0.493197i
$926$	19.1244	+	5.12436i	0.628465	+	0.168397i
$927$	−6.00000	+	10.3923i	−0.197066	+	0.341328i
$928$	48.4974	+	28.0000i	1.59201	+	0.919145i
$929$	−25.0000	−	43.3013i	−0.820223	−	1.42067i	−0.905516	−	0.424313i	$-0.860516\pi$
$929$	−25.0000	−	43.3013i	−0.820223	−	1.42067i	0.0852924	−	0.996356i	$-0.472818\pi$
$930$		0			0
$931$		0			0
$932$	−32.0000			−1.04819
$933$		0			0
$934$	31.1769	+	18.0000i	1.02014	+	0.588978i
$935$	6.92820	+	4.00000i	0.226576	+	0.130814i
$936$		0			0
$937$			58.0000i			1.89478i	0.320085	+	0.947389i	$0.396288\pi$
$937$			58.0000i			1.89478i	−0.320085	+	0.947389i	$0.603712\pi$
$938$		0			0
$939$		0			0
$940$	65.5692	+	17.5692i	2.13863	+	0.573045i
$941$	−9.51666	−	35.5167i	−0.310234	−	1.15781i	−0.928345	−	0.371719i	$-0.878768\pi$
$941$	−9.51666	−	35.5167i	−0.310234	−	1.15781i	0.618111	−	0.786091i	$-0.287898\pi$
$942$		0			0
$943$	30.0000	+	51.9615i	0.976934	+	1.69210i
$944$	32.0000	+	32.0000i	1.04151	+	1.04151i
$945$		0			0
$946$	2.00000	+	2.00000i	0.0650256	+	0.0650256i
$947$	−45.0788	−	12.0788i	−1.46487	−	0.392510i	−0.563698	−	0.825981i	$-0.690622\pi$
$947$	−45.0788	−	12.0788i	−1.46487	−	0.392510i	−0.901167	+	0.433471i	$0.857289\pi$
$948$		0			0
$949$		0			0
$950$	−6.00000	−	10.3923i	−0.194666	−	0.337171i
$951$		0			0
$952$		0			0
$953$		−	16.0000i		−	0.518291i	−0.965838	−	0.259145i	$-0.916559\pi$
$953$		−	16.0000i		−	0.518291i	0.965838	−	0.259145i	$-0.0834409\pi$
$954$	5.19615	−	3.00000i	0.168232	−	0.0971286i
$955$	−49.1769	+	13.1769i	−1.59133	+	0.426395i
$956$		0			0
$957$		0			0
$958$	−20.0000	+	20.0000i	−0.646171	+	0.646171i
$959$		0			0
$960$		0			0
$961$	−16.5000	−	28.5788i	−0.532258	−	0.921898i
$962$		0			0
$963$	−5.49038	−	20.4904i	−0.176925	−	0.660293i
$964$	38.1051	−	22.0000i	1.22728	−	0.708572i
$965$	32.0000	+	32.0000i	1.03012	+	1.03012i
$966$		0			0
$967$			2.00000i			0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$			2.00000i			0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	−24.5885	−	6.58846i	−0.790303	−	0.211761i
$969$		0			0
$970$	−4.00000	+	6.92820i	−0.128432	+	0.222451i
$971$	1.46410	−	5.46410i	0.0469853	−	0.175351i	−0.938446	−	0.345426i	$-0.887734\pi$
$971$	1.46410	−	5.46410i	0.0469853	−	0.175351i	0.985431	+	0.170075i	$0.0544010\pi$
$972$		0			0
$973$		0			0
$974$	−22.0000	−	22.0000i	−0.704925	−	0.704925i
$975$		0			0
$976$	8.78461	+	32.7846i	0.281189	+	1.04941i
$977$	−19.0000	+	32.9090i	−0.607864	+	1.05285i	0.383728	+	0.923446i	$0.374640\pi$
$977$	−19.0000	+	32.9090i	−0.607864	+	1.05285i	−0.991592	+	0.129405i	$0.958693\pi$
$978$		0			0
$979$	−14.0000	+	14.0000i	−0.447442	+	0.447442i
$980$		0			0
$981$	−9.00000	−	9.00000i	−0.287348	−	0.287348i
$982$	−19.0000	−	32.9090i	−0.606314	−	1.05017i
$983$	38.1051	+	22.0000i	1.21536	+	0.701691i	0.963923	−	0.266181i	$-0.0857620\pi$
$983$	38.1051	+	22.0000i	1.21536	+	0.701691i	0.251442	+	0.967872i	$0.419095\pi$
$984$		0			0
$985$	−17.3205	+	10.0000i	−0.551877	+	0.318626i
$986$			28.0000i			0.891702i
$987$		0			0
$988$		0			0
$989$	8.19615	+	2.19615i	0.260622	+	0.0698336i
$990$	−4.39230	−	16.3923i	−0.139597	−	0.520982i
$991$	−11.0000	+	19.0526i	−0.349427	+	0.605224i	−0.986148	−	0.165870i	$-0.946957\pi$
$991$	−11.0000	+	19.0526i	−0.349427	+	0.605224i	0.636721	+	0.771094i	$0.280290\pi$
$992$	−43.7128	−	11.7128i	−1.38788	−	0.371882i
$993$		0			0
$994$		0			0
$995$	48.0000	−	48.0000i	1.52170	−	1.52170i
$996$		0			0
$997$	−40.9808	+	10.9808i	−1.29787	+	0.347764i	−0.840646	−	0.541585i	$-0.817824\pi$
$997$	−40.9808	+	10.9808i	−1.29787	+	0.347764i	−0.457228	+	0.889350i	$0.651158\pi$
$998$	39.8372	+	23.0000i	1.26102	+	0.728052i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.e.165.1		4
7.2	even	3	inner	784.2.x.e.373.1		4
7.3	odd	6		112.2.m.b.85.1	yes	2
7.4	even	3		784.2.m.a.197.1		2
7.5	odd	6		784.2.x.d.373.1		4
7.6	odd	2		784.2.x.d.165.1		4
16.13	even	4	inner	784.2.x.e.557.1		4
28.3	even	6		448.2.m.a.113.1		2
56.3	even	6		896.2.m.c.225.1		2
56.45	odd	6		896.2.m.b.225.1		2
112.3	even	12		448.2.m.a.337.1		2
112.13	odd	4		784.2.x.d.557.1		4
112.45	odd	12		112.2.m.b.29.1	✓	2
112.59	even	12		896.2.m.c.673.1		2
112.61	odd	12		784.2.x.d.765.1		4
112.93	even	12	inner	784.2.x.e.765.1		4
112.101	odd	12		896.2.m.b.673.1		2
112.109	even	12		784.2.m.a.589.1		2
224.3	even	24		7168.2.a.k.1.1		2
224.45	odd	24		7168.2.a.b.1.2		2
224.115	even	24		7168.2.a.k.1.2		2
224.157	odd	24		7168.2.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.b.29.1	✓	2	112.45	odd	12
112.2.m.b.85.1	yes	2	7.3	odd	6
448.2.m.a.113.1		2	28.3	even	6
448.2.m.a.337.1		2	112.3	even	12
784.2.m.a.197.1		2	7.4	even	3
784.2.m.a.589.1		2	112.109	even	12
784.2.x.d.165.1		4	7.6	odd	2
784.2.x.d.373.1		4	7.5	odd	6
784.2.x.d.557.1		4	112.13	odd	4
784.2.x.d.765.1		4	112.61	odd	12
784.2.x.e.165.1		4	1.1	even	1	trivial
784.2.x.e.373.1		4	7.2	even	3	inner
784.2.x.e.557.1		4	16.13	even	4	inner
784.2.x.e.765.1		4	112.93	even	12	inner
896.2.m.b.225.1		2	56.45	odd	6
896.2.m.b.673.1		2	112.101	odd	12
896.2.m.c.225.1		2	56.3	even	6
896.2.m.c.673.1		2	112.59	even	12
7168.2.a.b.1.1		2	224.157	odd	24
7168.2.a.b.1.2		2	224.45	odd	24
7168.2.a.k.1.1		2	224.3	even	24
7168.2.a.k.1.2		2	224.115	even	24

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.x (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(-0.732051 + 2.73205i) q^{5} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)	\(q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(-0.732051 + 2.73205i) q^{5} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +(-2.00000 + 3.46410i) q^{10} +(-1.36603 + 0.366025i) q^{11} +(2.00000 + 3.46410i) q^{16} +(-1.00000 + 1.73205i) q^{17} +(4.09808 - 1.09808i) q^{18} +(2.73205 + 0.732051i) q^{19} +(-4.00000 + 4.00000i) q^{20} -2.00000 q^{22} +(-5.19615 + 3.00000i) q^{23} +(-2.59808 - 1.50000i) q^{25} +(7.00000 - 7.00000i) q^{29} +(-4.00000 + 6.92820i) q^{31} +(1.46410 + 5.46410i) q^{32} +(-2.00000 + 2.00000i) q^{34} +6.00000 q^{36} +(-1.83013 + 6.83013i) q^{37} +(3.46410 + 2.00000i) q^{38} +(-6.92820 + 4.00000i) q^{40} -10.0000i q^{41} +(-1.00000 - 1.00000i) q^{43} +(-2.73205 - 0.732051i) q^{44} +(2.19615 + 8.19615i) q^{45} +(-8.19615 + 2.19615i) q^{46} +(-6.00000 - 10.3923i) q^{47} +(-3.00000 - 3.00000i) q^{50} +(1.36603 - 0.366025i) q^{53} -4.00000i q^{55} +(12.1244 - 7.00000i) q^{58} +(10.9282 - 2.92820i) q^{59} +(8.19615 + 2.19615i) q^{61} +(-8.00000 + 8.00000i) q^{62} +8.00000i q^{64} +(1.09808 + 4.09808i) q^{67} +(-3.46410 + 2.00000i) q^{68} +(8.19615 + 2.19615i) q^{72} +(-5.19615 - 3.00000i) q^{73} +(-5.00000 + 8.66025i) q^{74} +(4.00000 + 4.00000i) q^{76} +(-5.00000 - 8.66025i) q^{79} +(-10.9282 + 2.92820i) q^{80} +(4.50000 - 7.79423i) q^{81} +(3.66025 - 13.6603i) q^{82} +(10.0000 - 10.0000i) q^{83} +(-4.00000 - 4.00000i) q^{85} +(-1.00000 - 1.73205i) q^{86} +(-3.46410 - 2.00000i) q^{88} +(12.1244 - 7.00000i) q^{89} +12.0000i q^{90} -12.0000 q^{92} +(-4.39230 - 16.3923i) q^{94} +(-4.00000 + 6.92820i) q^{95} +2.00000 q^{97} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{5} + 8 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^5 + 8 * q^8	\( 4 q + 2 q^{2} + 4 q^{5} + 8 q^{8} - 8 q^{10} - 2 q^{11} + 8 q^{16} - 4 q^{17} + 6 q^{18} + 4 q^{19} - 16 q^{20} - 8 q^{22} + 28 q^{29} - 16 q^{31} - 8 q^{32} - 8 q^{34} + 24 q^{36} + 10 q^{37} - 4 q^{43} - 4 q^{44} - 12 q^{45} - 12 q^{46} - 24 q^{47} - 12 q^{50} + 2 q^{53} + 16 q^{59} + 12 q^{61} - 32 q^{62} - 6 q^{67} + 12 q^{72} - 20 q^{74} + 16 q^{76} - 20 q^{79} - 16 q^{80} + 18 q^{81} - 20 q^{82} + 40 q^{83} - 16 q^{85} - 4 q^{86} - 48 q^{92} + 24 q^{94} - 16 q^{95} + 8 q^{97} - 12 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^5 + 8 * q^8 - 8 * q^10 - 2 * q^11 + 8 * q^16 - 4 * q^17 + 6 * q^18 + 4 * q^19 - 16 * q^20 - 8 * q^22 + 28 * q^29 - 16 * q^31 - 8 * q^32 - 8 * q^34 + 24 * q^36 + 10 * q^37 - 4 * q^43 - 4 * q^44 - 12 * q^45 - 12 * q^46 - 24 * q^47 - 12 * q^50 + 2 * q^53 + 16 * q^59 + 12 * q^61 - 32 * q^62 - 6 * q^67 + 12 * q^72 - 20 * q^74 + 16 * q^76 - 20 * q^79 - 16 * q^80 + 18 * q^81 - 20 * q^82 + 40 * q^83 - 16 * q^85 - 4 * q^86 - 48 * q^92 + 24 * q^94 - 16 * q^95 + 8 * q^97 - 12 * q^99

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