LMFDB - Embedded newform 112.2.m.b.85.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,2,Mod(29,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("112.29");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	112.m (of order $4$, 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:	$0.894324502638$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			85.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	112.85
Dual form			112.2.m.b.29.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(2.00000 + 2.00000i) q^{5} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(2.00000 + 2.00000i) q^{5} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} -4.00000 q^{10} +(1.00000 + 1.00000i) q^{11} +(1.00000 + 1.00000i) q^{14} -4.00000 q^{16} -2.00000 q^{17} +(-3.00000 - 3.00000i) q^{18} +(2.00000 - 2.00000i) q^{19} +(4.00000 - 4.00000i) q^{20} -2.00000 q^{22} -6.00000i q^{23} +3.00000i q^{25} -2.00000 q^{28} +(7.00000 - 7.00000i) q^{29} -8.00000 q^{31} +(4.00000 - 4.00000i) q^{32} +(2.00000 - 2.00000i) q^{34} +(2.00000 - 2.00000i) q^{35} +6.00000 q^{36} +(-5.00000 - 5.00000i) q^{37} +4.00000i q^{38} +8.00000i q^{40} +10.0000i q^{41} +(-1.00000 - 1.00000i) q^{43} +(2.00000 - 2.00000i) q^{44} +(-6.00000 + 6.00000i) q^{45} +(6.00000 + 6.00000i) q^{46} -12.0000 q^{47} -1.00000 q^{49} +(-3.00000 - 3.00000i) q^{50} +(-1.00000 - 1.00000i) q^{53} +4.00000i q^{55} +(2.00000 - 2.00000i) q^{56} +14.0000i q^{58} +(8.00000 + 8.00000i) q^{59} +(6.00000 - 6.00000i) q^{61} +(8.00000 - 8.00000i) q^{62} +3.00000 q^{63} +8.00000i q^{64} +(3.00000 - 3.00000i) q^{67} +4.00000i q^{68} +4.00000i q^{70} +(-6.00000 + 6.00000i) q^{72} -6.00000i q^{73} +10.0000 q^{74} +(-4.00000 - 4.00000i) q^{76} +(1.00000 - 1.00000i) q^{77} +10.0000 q^{79} +(-8.00000 - 8.00000i) q^{80} -9.00000 q^{81} +(-10.0000 - 10.0000i) q^{82} +(-10.0000 + 10.0000i) q^{83} +(-4.00000 - 4.00000i) q^{85} +2.00000 q^{86} +4.00000i q^{88} -14.0000i q^{89} -12.0000i q^{90} -12.0000 q^{92} +(12.0000 - 12.0000i) q^{94} +8.00000 q^{95} -2.00000 q^{97} +(1.00000 - 1.00000i) q^{98} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{5} + 4 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 4 * q^5 + 4 * q^8	$ 2 q - 2 q^{2} + 4 q^{5} + 4 q^{8} - 8 q^{10} + 2 q^{11} + 2 q^{14} - 8 q^{16} - 4 q^{17} - 6 q^{18} + 4 q^{19} + 8 q^{20} - 4 q^{22} - 4 q^{28} + 14 q^{29} - 16 q^{31} + 8 q^{32} + 4 q^{34} + 4 q^{35} + 12 q^{36} - 10 q^{37} - 2 q^{43} + 4 q^{44} - 12 q^{45} + 12 q^{46} - 24 q^{47} - 2 q^{49} - 6 q^{50} - 2 q^{53} + 4 q^{56} + 16 q^{59} + 12 q^{61} + 16 q^{62} + 6 q^{63} + 6 q^{67} - 12 q^{72} + 20 q^{74} - 8 q^{76} + 2 q^{77} + 20 q^{79} - 16 q^{80} - 18 q^{81} - 20 q^{82} - 20 q^{83} - 8 q^{85} + 4 q^{86} - 24 q^{92} + 24 q^{94} + 16 q^{95} - 4 q^{97} + 2 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^5 + 4 * q^8 - 8 * q^10 + 2 * q^11 + 2 * q^14 - 8 * q^16 - 4 * q^17 - 6 * q^18 + 4 * q^19 + 8 * q^20 - 4 * q^22 - 4 * q^28 + 14 * q^29 - 16 * q^31 + 8 * q^32 + 4 * q^34 + 4 * q^35 + 12 * q^36 - 10 * q^37 - 2 * q^43 + 4 * q^44 - 12 * q^45 + 12 * q^46 - 24 * q^47 - 2 * q^49 - 6 * q^50 - 2 * q^53 + 4 * q^56 + 16 * q^59 + 12 * q^61 + 16 * q^62 + 6 * q^63 + 6 * q^67 - 12 * q^72 + 20 * q^74 - 8 * q^76 + 2 * q^77 + 20 * q^79 - 16 * q^80 - 18 * q^81 - 20 * q^82 - 20 * q^83 - 8 * q^85 + 4 * q^86 - 24 * q^92 + 24 * q^94 + 16 * q^95 - 4 * q^97 + 2 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	2.00000i		−	1.00000i
$5$	2.00000	+	2.00000i	0.894427	+	0.894427i	0.994936	−	0.100509i	$-0.0320471\pi$
$5$	2.00000	+	2.00000i	0.894427	+	0.894427i	−0.100509	+	0.994936i	$0.532047\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$			3.00000i			1.00000i
$10$	−4.00000			−1.26491
$11$	1.00000	+	1.00000i	0.301511	+	0.301511i	0.841605	−	0.540094i	$-0.181611\pi$
$11$	1.00000	+	1.00000i	0.301511	+	0.301511i	−0.540094	+	0.841605i	$0.681611\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	1.00000	+	1.00000i	0.267261	+	0.267261i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$	−3.00000	−	3.00000i	−0.707107	−	0.707107i
$19$	2.00000	−	2.00000i	0.458831	−	0.458831i	−0.439440	−	0.898272i	$-0.644823\pi$
$19$	2.00000	−	2.00000i	0.458831	−	0.458831i	0.898272	+	0.439440i	$0.144823\pi$
$20$	4.00000	−	4.00000i	0.894427	−	0.894427i
$21$		0			0
$22$	−2.00000			−0.426401
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$			3.00000i			0.600000i
$26$		0			0
$27$		0			0
$28$	−2.00000			−0.377964
$29$	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
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$		0			0
$34$	2.00000	−	2.00000i	0.342997	−	0.342997i
$35$	2.00000	−	2.00000i	0.338062	−	0.338062i
$36$	6.00000			1.00000
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	−0.986394	+	0.164399i	$0.947432\pi$
$38$			4.00000i			0.648886i
$39$		0			0
$40$			8.00000i			1.26491i
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		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	$0.784391\pi$
$44$	2.00000	−	2.00000i	0.301511	−	0.301511i
$45$	−6.00000	+	6.00000i	−0.894427	+	0.894427i
$46$	6.00000	+	6.00000i	0.884652	+	0.884652i
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	−3.00000	−	3.00000i	−0.424264	−	0.424264i
$51$		0			0
$52$		0			0
$53$	−1.00000	−	1.00000i	−0.137361	−	0.137361i	0.635083	−	0.772444i	$-0.280966\pi$
$53$	−1.00000	−	1.00000i	−0.137361	−	0.137361i	−0.772444	+	0.635083i	$0.780966\pi$
$54$		0			0
$55$			4.00000i			0.539360i
$56$	2.00000	−	2.00000i	0.267261	−	0.267261i
$57$		0			0
$58$			14.0000i			1.83829i
$59$	8.00000	+	8.00000i	1.04151	+	1.04151i	0.999100	+	0.0424110i	$0.0135039\pi$
$59$	8.00000	+	8.00000i	1.04151	+	1.04151i	0.0424110	+	0.999100i	$0.486496\pi$
$60$		0			0
$61$	6.00000	−	6.00000i	0.768221	−	0.768221i	−0.209572	−	0.977793i	$-0.567207\pi$
$61$	6.00000	−	6.00000i	0.768221	−	0.768221i	0.977793	+	0.209572i	$0.0672070\pi$
$62$	8.00000	−	8.00000i	1.01600	−	1.01600i
$63$	3.00000			0.377964
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	3.00000	−	3.00000i	0.366508	−	0.366508i	−0.499694	−	0.866202i	$-0.666554\pi$
$67$	3.00000	−	3.00000i	0.366508	−	0.366508i	0.866202	+	0.499694i	$0.166554\pi$
$68$			4.00000i			0.485071i
$69$		0			0
$70$			4.00000i			0.478091i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−6.00000	+	6.00000i	−0.707107	+	0.707107i
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	10.0000			1.16248
$75$		0			0
$76$	−4.00000	−	4.00000i	−0.458831	−	0.458831i
$77$	1.00000	−	1.00000i	0.113961	−	0.113961i
$78$		0			0
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$	−8.00000	−	8.00000i	−0.894427	−	0.894427i
$81$	−9.00000			−1.00000
$82$	−10.0000	−	10.0000i	−1.10432	−	1.10432i
$83$	−10.0000	+	10.0000i	−1.09764	+	1.09764i	−0.102957	+	0.994686i	$0.532830\pi$
$83$	−10.0000	+	10.0000i	−1.09764	+	1.09764i	−0.994686	+	0.102957i	$0.967170\pi$
$84$		0			0
$85$	−4.00000	−	4.00000i	−0.433861	−	0.433861i
$86$	2.00000			0.215666
$87$		0			0
$88$			4.00000i			0.426401i
$89$		−	14.0000i		−	1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$		−	14.0000i		−	1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$		−	12.0000i		−	1.26491i
$91$		0			0
$92$	−12.0000			−1.25109
$93$		0			0
$94$	12.0000	−	12.0000i	1.23771	−	1.23771i
$95$	8.00000			0.820783
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$	1.00000	−	1.00000i	0.101015	−	0.101015i
$99$	−3.00000	+	3.00000i	−0.301511	+	0.301511i
$100$	6.00000			0.600000
$101$	6.00000	+	6.00000i	0.597022	+	0.597022i	0.939519	−	0.342497i	$-0.111273\pi$
$101$	6.00000	+	6.00000i	0.597022	+	0.597022i	−0.342497	+	0.939519i	$0.611273\pi$
$102$		0			0
$103$			4.00000i			0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$			4.00000i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$		0			0
$105$		0			0
$106$	2.00000			0.194257
$107$	5.00000	+	5.00000i	0.483368	+	0.483368i	0.906206	−	0.422837i	$-0.138966\pi$
$107$	5.00000	+	5.00000i	0.483368	+	0.483368i	−0.422837	+	0.906206i	$0.638966\pi$
$108$		0			0
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	−0.836031	−	0.548683i	$-0.815129\pi$
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	0.548683	+	0.836031i	$0.315129\pi$
$110$	−4.00000	−	4.00000i	−0.381385	−	0.381385i
$111$		0			0
$112$			4.00000i			0.377964i
$113$	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
$115$	12.0000	−	12.0000i	1.11901	−	1.11901i
$116$	−14.0000	−	14.0000i	−1.29987	−	1.29987i
$117$		0			0
$118$	−16.0000			−1.47292
$119$			2.00000i			0.183340i
$120$		0			0
$121$		−	9.00000i		−	0.818182i
$122$			12.0000i			1.08643i
$123$		0			0
$124$			16.0000i			1.43684i
$125$	4.00000	−	4.00000i	0.357771	−	0.357771i
$126$	−3.00000	+	3.00000i	−0.267261	+	0.267261i
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−8.00000	−	8.00000i	−0.707107	−	0.707107i
$129$		0			0
$130$		0			0
$131$	−14.0000	+	14.0000i	−1.22319	+	1.22319i	−0.256693	+	0.966493i	$0.582633\pi$
$131$	−14.0000	+	14.0000i	−1.22319	+	1.22319i	−0.966493	+	0.256693i	$0.917367\pi$
$132$		0			0
$133$	−2.00000	−	2.00000i	−0.173422	−	0.173422i
$134$			6.00000i			0.518321i
$135$		0			0
$136$	−4.00000	−	4.00000i	−0.342997	−	0.342997i
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	−2.00000	−	2.00000i	−0.169638	−	0.169638i	0.617182	−	0.786820i	$-0.288274\pi$
$139$	−2.00000	−	2.00000i	−0.169638	−	0.169638i	−0.786820	+	0.617182i	$0.788274\pi$
$140$	−4.00000	−	4.00000i	−0.338062	−	0.338062i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	12.0000i		−	1.00000i
$145$	28.0000			2.32527
$146$	6.00000	+	6.00000i	0.496564	+	0.496564i
$147$		0			0
$148$	−10.0000	+	10.0000i	−0.821995	+	0.821995i
$149$	3.00000	+	3.00000i	0.245770	+	0.245770i	0.819232	−	0.573462i	$-0.194400\pi$
$149$	3.00000	+	3.00000i	0.245770	+	0.245770i	−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$	8.00000			0.648886
$153$		−	6.00000i		−	0.485071i
$154$			2.00000i			0.161165i
$155$	−16.0000	−	16.0000i	−1.28515	−	1.28515i
$156$		0			0
$157$	−12.0000	+	12.0000i	−0.957704	+	0.957704i	−0.999141	−	0.0414369i	$-0.986806\pi$
$157$	−12.0000	+	12.0000i	−0.957704	+	0.957704i	0.0414369	+	0.999141i	$0.486806\pi$
$158$	−10.0000	+	10.0000i	−0.795557	+	0.795557i
$159$		0			0
$160$	16.0000			1.26491
$161$	−6.00000			−0.472866
$162$	9.00000	−	9.00000i	0.707107	−	0.707107i
$163$	15.0000	−	15.0000i	1.17489	−	1.17489i	0.193862	−	0.981029i	$-0.437899\pi$
$163$	15.0000	−	15.0000i	1.17489	−	1.17489i	0.981029	−	0.193862i	$-0.0621013\pi$
$164$	20.0000			1.56174
$165$		0			0
$166$		−	20.0000i		−	1.55230i
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$	8.00000			0.613572
$171$	6.00000	+	6.00000i	0.458831	+	0.458831i
$172$	−2.00000	+	2.00000i	−0.152499	+	0.152499i
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$	3.00000			0.226779
$176$	−4.00000	−	4.00000i	−0.301511	−	0.301511i
$177$		0			0
$178$	14.0000	+	14.0000i	1.04934	+	1.04934i
$179$	−3.00000	+	3.00000i	−0.224231	+	0.224231i	−0.810277	−	0.586047i	$-0.800683\pi$
$179$	−3.00000	+	3.00000i	−0.224231	+	0.224231i	0.586047	+	0.810277i	$0.300683\pi$
$180$	12.0000	+	12.0000i	0.894427	+	0.894427i
$181$	−4.00000	−	4.00000i	−0.297318	−	0.297318i	0.542645	−	0.839962i	$-0.317423\pi$
$181$	−4.00000	−	4.00000i	−0.297318	−	0.297318i	−0.839962	+	0.542645i	$0.817423\pi$
$182$		0			0
$183$		0			0
$184$	12.0000	−	12.0000i	0.884652	−	0.884652i
$185$		−	20.0000i		−	1.47043i
$186$		0			0
$187$	−2.00000	−	2.00000i	−0.146254	−	0.146254i
$188$			24.0000i			1.75038i
$189$		0			0
$190$	−8.00000	+	8.00000i	−0.580381	+	0.580381i
$191$	−18.0000			−1.30243			−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000			−1.30243			−0.651217	+	0.758891i	$0.725741\pi$
$192$		0			0
$193$	−16.0000			−1.15171			−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000			−1.15171			−0.575853	+	0.817554i	$0.695330\pi$
$194$	2.00000	−	2.00000i	0.143592	−	0.143592i
$195$		0			0
$196$			2.00000i			0.142857i
$197$	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	$0.668946\pi$
$198$		−	6.00000i		−	0.426401i
$199$		−	24.0000i		−	1.70131i	−0.525720	−	0.850657i	$-0.676204\pi$
$199$		−	24.0000i		−	1.70131i	0.525720	−	0.850657i	$-0.323796\pi$
$200$	−6.00000	+	6.00000i	−0.424264	+	0.424264i
$201$		0			0
$202$	−12.0000			−0.844317
$203$	−7.00000	−	7.00000i	−0.491304	−	0.491304i
$204$		0			0
$205$	−20.0000	+	20.0000i	−1.39686	+	1.39686i
$206$	−4.00000	−	4.00000i	−0.278693	−	0.278693i
$207$	18.0000			1.25109
$208$		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	$0.394353\pi$
$212$	−2.00000	+	2.00000i	−0.137361	+	0.137361i
$213$		0			0
$214$	−10.0000			−0.683586
$215$		−	4.00000i		−	0.272798i
$216$		0			0
$217$			8.00000i			0.543075i
$218$		−	6.00000i		−	0.406371i
$219$		0			0
$220$	8.00000			0.539360
$221$		0			0
$222$		0			0
$223$	4.00000			0.267860			0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000			0.267860			0.133930	+	0.990991i	$0.457240\pi$
$224$	−4.00000	−	4.00000i	−0.267261	−	0.267261i
$225$	−9.00000			−0.600000
$226$	−4.00000	+	4.00000i	−0.266076	+	0.266076i
$227$	−2.00000	+	2.00000i	−0.132745	+	0.132745i	−0.770357	−	0.637613i	$-0.779922\pi$
$227$	−2.00000	+	2.00000i	−0.132745	+	0.132745i	0.637613	+	0.770357i	$0.279922\pi$
$228$		0			0
$229$	8.00000	+	8.00000i	0.528655	+	0.528655i	0.920171	−	0.391516i	$-0.128049\pi$
$229$	8.00000	+	8.00000i	0.528655	+	0.528655i	−0.391516	+	0.920171i	$0.628049\pi$
$230$			24.0000i			1.58251i
$231$		0			0
$232$	28.0000			1.83829
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$		0			0
$235$	−24.0000	−	24.0000i	−1.56559	−	1.56559i
$236$	16.0000	−	16.0000i	1.04151	−	1.04151i
$237$		0			0
$238$	−2.00000	−	2.00000i	−0.129641	−	0.129641i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$	9.00000	+	9.00000i	0.578542	+	0.578542i
$243$		0			0
$244$	−12.0000	−	12.0000i	−0.768221	−	0.768221i
$245$	−2.00000	−	2.00000i	−0.127775	−	0.127775i
$246$		0			0
$247$		0			0
$248$	−16.0000	−	16.0000i	−1.01600	−	1.01600i
$249$		0			0
$250$			8.00000i			0.505964i
$251$	−14.0000	−	14.0000i	−0.883672	−	0.883672i	0.110234	−	0.993906i	$-0.464840\pi$
$251$	−14.0000	−	14.0000i	−0.883672	−	0.883672i	−0.993906	+	0.110234i	$0.964840\pi$
$252$		−	6.00000i		−	0.377964i
$253$	6.00000	−	6.00000i	0.377217	−	0.377217i
$254$	−8.00000	+	8.00000i	−0.501965	+	0.501965i
$255$		0			0
$256$	16.0000			1.00000
$257$	−2.00000			−0.124757			−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000			−0.124757			−0.0623783	+	0.998053i	$0.519869\pi$
$258$		0			0
$259$	−5.00000	+	5.00000i	−0.310685	+	0.310685i
$260$		0			0
$261$	21.0000	+	21.0000i	1.29987	+	1.29987i
$262$		−	28.0000i		−	1.72985i
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$		−	4.00000i		−	0.245718i
$266$	4.00000			0.245256
$267$		0			0
$268$	−6.00000	−	6.00000i	−0.366508	−	0.366508i
$269$	−8.00000	+	8.00000i	−0.487769	+	0.487769i	−0.907601	−	0.419833i	$-0.862089\pi$
$269$	−8.00000	+	8.00000i	−0.487769	+	0.487769i	0.419833	+	0.907601i	$0.362089\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$	8.00000			0.485071
$273$		0			0
$274$	−12.0000	−	12.0000i	−0.724947	−	0.724947i
$275$	−3.00000	+	3.00000i	−0.180907	+	0.180907i
$276$		0			0
$277$	−15.0000	−	15.0000i	−0.901263	−	0.901263i	0.0942828	−	0.995545i	$-0.469944\pi$
$277$	−15.0000	−	15.0000i	−0.901263	−	0.901263i	−0.995545	+	0.0942828i	$0.969944\pi$
$278$	4.00000			0.239904
$279$		−	24.0000i		−	1.43684i
$280$	8.00000			0.478091
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	4.00000	+	4.00000i	0.237775	+	0.237775i	0.815928	−	0.578153i	$-0.196226\pi$
$283$	4.00000	+	4.00000i	0.237775	+	0.237775i	−0.578153	+	0.815928i	$0.696226\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.0000			0.590281
$288$	12.0000	+	12.0000i	0.707107	+	0.707107i
$289$	−13.0000			−0.764706
$290$	−28.0000	+	28.0000i	−1.64422	+	1.64422i
$291$		0			0
$292$	−12.0000			−0.702247
$293$	14.0000	+	14.0000i	0.817889	+	0.817889i	0.985802	−	0.167913i	$-0.0537028\pi$
$293$	14.0000	+	14.0000i	0.817889	+	0.817889i	−0.167913	+	0.985802i	$0.553703\pi$
$294$		0			0
$295$			32.0000i			1.86311i
$296$		−	20.0000i		−	1.16248i
$297$		0			0
$298$	−6.00000			−0.347571
$299$		0			0
$300$		0			0
$301$	−1.00000	+	1.00000i	−0.0576390	+	0.0576390i
$302$	−10.0000	−	10.0000i	−0.575435	−	0.575435i
$303$		0			0
$304$	−8.00000	+	8.00000i	−0.458831	+	0.458831i
$305$	24.0000			1.37424
$306$	6.00000	+	6.00000i	0.342997	+	0.342997i
$307$	18.0000	−	18.0000i	1.02731	−	1.02731i	0.0276979	−	0.999616i	$-0.491182\pi$
$307$	18.0000	−	18.0000i	1.02731	−	1.02731i	0.999616	−	0.0276979i	$-0.00881765\pi$
$308$	−2.00000	−	2.00000i	−0.113961	−	0.113961i
$309$		0			0
$310$	32.0000			1.81748
$311$			20.0000i			1.13410i	0.823685	+	0.567048i	$0.191915\pi$
$311$			20.0000i			1.13410i	−0.823685	+	0.567048i	$0.808085\pi$
$312$		0			0
$313$			14.0000i			0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$			14.0000i			0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$		−	24.0000i		−	1.35440i
$315$	6.00000	+	6.00000i	0.338062	+	0.338062i
$316$		−	20.0000i		−	1.12509i
$317$	−7.00000	+	7.00000i	−0.393159	+	0.393159i	−0.875812	−	0.482653i	$-0.839673\pi$
$317$	−7.00000	+	7.00000i	−0.393159	+	0.393159i	0.482653	+	0.875812i	$0.339673\pi$
$318$		0			0
$319$	14.0000			0.783850
$320$	−16.0000	+	16.0000i	−0.894427	+	0.894427i
$321$		0			0
$322$	6.00000	−	6.00000i	0.334367	−	0.334367i
$323$	−4.00000	+	4.00000i	−0.222566	+	0.222566i
$324$			18.0000i			1.00000i
$325$		0			0
$326$			30.0000i			1.66155i
$327$		0			0
$328$	−20.0000	+	20.0000i	−1.10432	+	1.10432i
$329$			12.0000i			0.661581i
$330$		0			0
$331$	21.0000	+	21.0000i	1.15426	+	1.15426i	0.985689	+	0.168576i	$0.0539168\pi$
$331$	21.0000	+	21.0000i	1.15426	+	1.15426i	0.168576	+	0.985689i	$0.446083\pi$
$332$	20.0000	+	20.0000i	1.09764	+	1.09764i
$333$	15.0000	−	15.0000i	0.821995	−	0.821995i
$334$	−12.0000	−	12.0000i	−0.656611	−	0.656611i
$335$	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	$0.430081\pi$
$338$	−13.0000	−	13.0000i	−0.707107	−	0.707107i
$339$		0			0
$340$	−8.00000	+	8.00000i	−0.433861	+	0.433861i
$341$	−8.00000	−	8.00000i	−0.433224	−	0.433224i
$342$	−12.0000			−0.648886
$343$			1.00000i			0.0539949i
$344$		−	4.00000i		−	0.215666i
$345$		0			0
$346$		0			0
$347$	15.0000	+	15.0000i	0.805242	+	0.805242i	0.983910	−	0.178667i	$-0.0571786\pi$
$347$	15.0000	+	15.0000i	0.805242	+	0.805242i	−0.178667	+	0.983910i	$0.557179\pi$
$348$		0			0
$349$	12.0000	−	12.0000i	0.642345	−	0.642345i	−0.308786	−	0.951131i	$-0.599923\pi$
$349$	12.0000	−	12.0000i	0.642345	−	0.642345i	0.951131	+	0.308786i	$0.0999228\pi$
$350$	−3.00000	+	3.00000i	−0.160357	+	0.160357i
$351$		0			0
$352$	8.00000			0.426401
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$		0			0
$356$	−28.0000			−1.48400
$357$		0			0
$358$		−	6.00000i		−	0.317110i
$359$			6.00000i			0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$			6.00000i			0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$	−24.0000			−1.26491
$361$			11.0000i			0.578947i
$362$	8.00000			0.420471
$363$		0			0
$364$		0			0
$365$	12.0000	−	12.0000i	0.628109	−	0.628109i
$366$		0			0
$367$	−32.0000			−1.67039			−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000			−1.67039			−0.835193	+	0.549957i	$0.814644\pi$
$368$			24.0000i			1.25109i
$369$	−30.0000			−1.56174
$370$	20.0000	+	20.0000i	1.03975	+	1.03975i
$371$	−1.00000	+	1.00000i	−0.0519174	+	0.0519174i
$372$		0			0
$373$	−11.0000	−	11.0000i	−0.569558	−	0.569558i	0.362446	−	0.932005i	$-0.381942\pi$
$373$	−11.0000	−	11.0000i	−0.569558	−	0.569558i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	4.00000			0.206835
$375$		0			0
$376$	−24.0000	−	24.0000i	−1.23771	−	1.23771i
$377$		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	$0.593467\pi$
$380$		−	16.0000i		−	0.820783i
$381$		0			0
$382$	18.0000	−	18.0000i	0.920960	−	0.920960i
$383$	4.00000			0.204390			0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000			0.204390			0.102195	+	0.994764i	$0.467413\pi$
$384$		0			0
$385$	4.00000			0.203859
$386$	16.0000	−	16.0000i	0.814379	−	0.814379i
$387$	3.00000	−	3.00000i	0.152499	−	0.152499i
$388$			4.00000i			0.203069i
$389$	3.00000	+	3.00000i	0.152106	+	0.152106i	0.779058	−	0.626952i	$-0.215698\pi$
$389$	3.00000	+	3.00000i	0.152106	+	0.152106i	−0.626952	+	0.779058i	$0.715698\pi$
$390$		0			0
$391$			12.0000i			0.606866i
$392$	−2.00000	−	2.00000i	−0.101015	−	0.101015i
$393$		0			0
$394$	−10.0000			−0.503793
$395$	20.0000	+	20.0000i	1.00631	+	1.00631i
$396$	6.00000	+	6.00000i	0.301511	+	0.301511i
$397$	−12.0000	+	12.0000i	−0.602263	+	0.602263i	−0.940913	−	0.338650i	$-0.890030\pi$
$397$	−12.0000	+	12.0000i	−0.602263	+	0.602263i	0.338650	+	0.940913i	$0.390030\pi$
$398$	24.0000	+	24.0000i	1.20301	+	1.20301i
$399$		0			0
$400$		−	12.0000i		−	0.600000i
$401$	12.0000			0.599251			0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000			0.599251			0.299626	+	0.954057i	$0.403138\pi$
$402$		0			0
$403$		0			0
$404$	12.0000	−	12.0000i	0.597022	−	0.597022i
$405$	−18.0000	−	18.0000i	−0.894427	−	0.894427i
$406$	14.0000			0.694808
$407$		−	10.0000i		−	0.495682i
$408$		0			0
$409$		−	14.0000i		−	0.692255i	−0.938187	−	0.346128i	$-0.887496\pi$
$409$		−	14.0000i		−	0.692255i	0.938187	−	0.346128i	$-0.112504\pi$
$410$		−	40.0000i		−	1.97546i
$411$		0			0
$412$	8.00000			0.394132
$413$	8.00000	−	8.00000i	0.393654	−	0.393654i
$414$	−18.0000	+	18.0000i	−0.884652	+	0.884652i
$415$	−40.0000			−1.96352
$416$		0			0
$417$		0			0
$418$	−4.00000	+	4.00000i	−0.195646	+	0.195646i
$419$	12.0000	−	12.0000i	0.586238	−	0.586238i	−0.350372	−	0.936611i	$-0.613945\pi$
$419$	12.0000	−	12.0000i	0.586238	−	0.586238i	0.936611	+	0.350372i	$0.113945\pi$
$420$		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	$0.850383\pi$
$422$		−	18.0000i		−	0.876226i
$423$		−	36.0000i		−	1.75038i
$424$		−	4.00000i		−	0.194257i
$425$		−	6.00000i		−	0.291043i
$426$		0			0
$427$	−6.00000	−	6.00000i	−0.290360	−	0.290360i
$428$	10.0000	−	10.0000i	0.483368	−	0.483368i
$429$		0			0
$430$	4.00000	+	4.00000i	0.192897	+	0.192897i
$431$	−8.00000			−0.385346			−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000			−0.385346			−0.192673	+	0.981263i	$0.561716\pi$
$432$		0			0
$433$	−6.00000			−0.288342			−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000			−0.288342			−0.144171	+	0.989553i	$0.546051\pi$
$434$	−8.00000	−	8.00000i	−0.384012	−	0.384012i
$435$		0			0
$436$	6.00000	+	6.00000i	0.287348	+	0.287348i
$437$	−12.0000	−	12.0000i	−0.574038	−	0.574038i
$438$		0			0
$439$			16.0000i			0.763638i	0.924237	+	0.381819i	$0.124702\pi$
$439$			16.0000i			0.763638i	−0.924237	+	0.381819i	$0.875298\pi$
$440$	−8.00000	+	8.00000i	−0.381385	+	0.381385i
$441$		−	3.00000i		−	0.142857i
$442$		0			0
$443$	−1.00000	−	1.00000i	−0.0475114	−	0.0475114i	0.682952	−	0.730463i	$-0.260696\pi$
$443$	−1.00000	−	1.00000i	−0.0475114	−	0.0475114i	−0.730463	+	0.682952i	$0.760696\pi$
$444$		0			0
$445$	28.0000	−	28.0000i	1.32733	−	1.32733i
$446$	−4.00000	+	4.00000i	−0.189405	+	0.189405i
$447$		0			0
$448$	8.00000			0.377964
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	9.00000	−	9.00000i	0.424264	−	0.424264i
$451$	−10.0000	+	10.0000i	−0.470882	+	0.470882i
$452$		−	8.00000i		−	0.376288i
$453$		0			0
$454$		−	4.00000i		−	0.187729i
$455$		0			0
$456$		0			0
$457$			22.0000i			1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$			22.0000i			1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	−16.0000			−0.747631
$459$		0			0
$460$	−24.0000	−	24.0000i	−1.11901	−	1.11901i
$461$	16.0000	−	16.0000i	0.745194	−	0.745194i	−0.228378	−	0.973572i	$-0.573342\pi$
$461$	16.0000	−	16.0000i	0.745194	−	0.745194i	0.973572	+	0.228378i	$0.0733423\pi$
$462$		0			0
$463$	14.0000			0.650635			0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000			0.650635			0.325318	+	0.945605i	$0.394529\pi$
$464$	−28.0000	+	28.0000i	−1.29987	+	1.29987i
$465$		0			0
$466$	16.0000	+	16.0000i	0.741186	+	0.741186i
$467$	18.0000	−	18.0000i	0.832941	−	0.832941i	−0.154977	−	0.987918i	$-0.549530\pi$
$467$	18.0000	−	18.0000i	0.832941	−	0.832941i	0.987918	+	0.154977i	$0.0495305\pi$
$468$		0			0
$469$	−3.00000	−	3.00000i	−0.138527	−	0.138527i
$470$	48.0000			2.21407
$471$		0			0
$472$			32.0000i			1.47292i
$473$		−	2.00000i		−	0.0919601i
$474$		0			0
$475$	6.00000	+	6.00000i	0.275299	+	0.275299i
$476$	4.00000			0.183340
$477$	3.00000	−	3.00000i	0.137361	−	0.137361i
$478$		0			0
$479$	−20.0000			−0.913823			−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000			−0.913823			−0.456912	+	0.889512i	$0.651044\pi$
$480$		0			0
$481$		0			0
$482$	−22.0000	+	22.0000i	−1.00207	+	1.00207i
$483$		0			0
$484$	−18.0000			−0.818182
$485$	−4.00000	−	4.00000i	−0.181631	−	0.181631i
$486$		0			0
$487$			22.0000i			0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$			22.0000i			0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$	24.0000			1.08643
$489$		0			0
$490$	4.00000			0.180702
$491$	−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
$493$	−14.0000	+	14.0000i	−0.630528	+	0.630528i
$494$		0			0
$495$	−12.0000			−0.539360
$496$	32.0000			1.43684
$497$		0			0
$498$		0			0
$499$	−23.0000	+	23.0000i	−1.02962	+	1.02962i	−0.0300737	+	0.999548i	$0.509574\pi$
$499$	−23.0000	+	23.0000i	−1.02962	+	1.02962i	−0.999548	+	0.0300737i	$0.990426\pi$
$500$	−8.00000	−	8.00000i	−0.357771	−	0.357771i
$501$		0			0
$502$	28.0000			1.24970
$503$		−	16.0000i		−	0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$		−	16.0000i		−	0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	6.00000	+	6.00000i	0.267261	+	0.267261i
$505$			24.0000i			1.06799i
$506$			12.0000i			0.533465i
$507$		0			0
$508$		−	16.0000i		−	0.709885i
$509$	−8.00000	+	8.00000i	−0.354594	+	0.354594i	−0.861816	−	0.507222i	$-0.830672\pi$
$509$	−8.00000	+	8.00000i	−0.354594	+	0.354594i	0.507222	+	0.861816i	$0.330672\pi$
$510$		0			0
$511$	−6.00000			−0.265424
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$		0			0
$514$	2.00000	−	2.00000i	0.0882162	−	0.0882162i
$515$	−8.00000	+	8.00000i	−0.352522	+	0.352522i
$516$		0			0
$517$	−12.0000	−	12.0000i	−0.527759	−	0.527759i
$518$		−	10.0000i		−	0.439375i
$519$		0			0
$520$		0			0
$521$		−	10.0000i		−	0.438108i	−0.975713	−	0.219054i	$-0.929703\pi$
$521$		−	10.0000i		−	0.438108i	0.975713	−	0.219054i	$-0.0702971\pi$
$522$	−42.0000			−1.83829
$523$	24.0000	+	24.0000i	1.04945	+	1.04945i	0.998712	+	0.0507346i	$0.0161562\pi$
$523$	24.0000	+	24.0000i	1.04945	+	1.04945i	0.0507346	+	0.998712i	$0.483844\pi$
$524$	28.0000	+	28.0000i	1.22319	+	1.22319i
$525$		0			0
$526$	16.0000	+	16.0000i	0.697633	+	0.697633i
$527$	16.0000			0.696971
$528$		0			0
$529$	−13.0000			−0.565217
$530$	4.00000	+	4.00000i	0.173749	+	0.173749i
$531$	−24.0000	+	24.0000i	−1.04151	+	1.04151i
$532$	−4.00000	+	4.00000i	−0.173422	+	0.173422i
$533$		0			0
$534$		0			0
$535$			20.0000i			0.864675i
$536$	12.0000			0.518321
$537$		0			0
$538$		−	16.0000i		−	0.689809i
$539$	−1.00000	−	1.00000i	−0.0430730	−	0.0430730i
$540$		0			0
$541$	11.0000	−	11.0000i	0.472927	−	0.472927i	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	11.0000	−	11.0000i	0.472927	−	0.472927i	0.902861	+	0.429934i	$0.141463\pi$
$542$	8.00000	−	8.00000i	0.343629	−	0.343629i
$543$		0			0
$544$	−8.00000	+	8.00000i	−0.342997	+	0.342997i
$545$	−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	$0.00523822\pi$
$548$	24.0000			1.02523
$549$	18.0000	+	18.0000i	0.768221	+	0.768221i
$550$		−	6.00000i		−	0.255841i
$551$		−	28.0000i		−	1.19284i
$552$		0			0
$553$		−	10.0000i		−	0.425243i
$554$	30.0000			1.27458
$555$		0			0
$556$	−4.00000	+	4.00000i	−0.169638	+	0.169638i
$557$	−17.0000	+	17.0000i	−0.720313	+	0.720313i	−0.968669	−	0.248356i	$-0.920110\pi$
$557$	−17.0000	+	17.0000i	−0.720313	+	0.720313i	0.248356	+	0.968669i	$0.420110\pi$
$558$	24.0000	+	24.0000i	1.01600	+	1.01600i
$559$		0			0
$560$	−8.00000	+	8.00000i	−0.338062	+	0.338062i
$561$		0			0
$562$		0			0
$563$	20.0000	−	20.0000i	0.842900	−	0.842900i	−0.146336	−	0.989235i	$-0.546748\pi$
$563$	20.0000	−	20.0000i	0.842900	−	0.842900i	0.989235	+	0.146336i	$0.0467479\pi$
$564$		0			0
$565$	8.00000	+	8.00000i	0.336563	+	0.336563i
$566$	−8.00000			−0.336265
$567$			9.00000i			0.377964i
$568$		0			0
$569$		−	14.0000i		−	0.586911i	−0.955973	−	0.293455i	$-0.905195\pi$
$569$		−	14.0000i		−	0.586911i	0.955973	−	0.293455i	$-0.0948052\pi$
$570$		0			0
$571$	−9.00000	−	9.00000i	−0.376638	−	0.376638i	0.493250	−	0.869888i	$-0.335809\pi$
$571$	−9.00000	−	9.00000i	−0.376638	−	0.376638i	−0.869888	+	0.493250i	$0.835809\pi$
$572$		0			0
$573$		0			0
$574$	−10.0000	+	10.0000i	−0.417392	+	0.417392i
$575$	18.0000			0.750652
$576$	−24.0000			−1.00000
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$	13.0000	−	13.0000i	0.540729	−	0.540729i
$579$		0			0
$580$		−	56.0000i		−	2.32527i
$581$	10.0000	+	10.0000i	0.414870	+	0.414870i
$582$		0			0
$583$		−	2.00000i		−	0.0828315i
$584$	12.0000	−	12.0000i	0.496564	−	0.496564i
$585$		0			0
$586$	−28.0000			−1.15667
$587$	10.0000	+	10.0000i	0.412744	+	0.412744i	0.882693	−	0.469949i	$-0.155728\pi$
$587$	10.0000	+	10.0000i	0.412744	+	0.412744i	−0.469949	+	0.882693i	$0.655728\pi$
$588$		0			0
$589$	−16.0000	+	16.0000i	−0.659269	+	0.659269i
$590$	−32.0000	−	32.0000i	−1.31742	−	1.31742i
$591$		0			0
$592$	20.0000	+	20.0000i	0.821995	+	0.821995i
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$	−4.00000	+	4.00000i	−0.163984	+	0.163984i
$596$	6.00000	−	6.00000i	0.245770	−	0.245770i
$597$		0			0
$598$		0			0
$599$		−	24.0000i		−	0.980613i	−0.871550	−	0.490307i	$-0.836885\pi$
$599$		−	24.0000i		−	0.980613i	0.871550	−	0.490307i	$-0.163115\pi$
$600$		0			0
$601$			10.0000i			0.407909i	0.978980	+	0.203954i	$0.0653794\pi$
$601$			10.0000i			0.407909i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		−	2.00000i		−	0.0815139i
$603$	9.00000	+	9.00000i	0.366508	+	0.366508i
$604$	20.0000			0.813788
$605$	18.0000	−	18.0000i	0.731804	−	0.731804i
$606$		0			0
$607$	28.0000			1.13648			0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000			1.13648			0.568242	+	0.822861i	$0.307624\pi$
$608$		−	16.0000i		−	0.648886i
$609$		0			0
$610$	−24.0000	+	24.0000i	−0.971732	+	0.971732i
$611$		0			0
$612$	−12.0000			−0.485071
$613$	9.00000	+	9.00000i	0.363507	+	0.363507i	0.865102	−	0.501596i	$-0.167253\pi$
$613$	9.00000	+	9.00000i	0.363507	+	0.363507i	−0.501596	+	0.865102i	$0.667253\pi$
$614$			36.0000i			1.45284i
$615$		0			0
$616$	4.00000			0.161165
$617$			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
$619$	−12.0000	−	12.0000i	−0.482321	−	0.482321i	0.423551	−	0.905872i	$-0.360783\pi$
$619$	−12.0000	−	12.0000i	−0.482321	−	0.482321i	−0.905872	+	0.423551i	$0.860783\pi$
$620$	−32.0000	+	32.0000i	−1.28515	+	1.28515i
$621$		0			0
$622$	−20.0000	−	20.0000i	−0.801927	−	0.801927i
$623$	−14.0000			−0.560898
$624$		0			0
$625$	31.0000			1.24000
$626$	−14.0000	−	14.0000i	−0.559553	−	0.559553i
$627$		0			0
$628$	24.0000	+	24.0000i	0.957704	+	0.957704i
$629$	10.0000	+	10.0000i	0.398726	+	0.398726i
$630$	−12.0000			−0.478091
$631$		−	40.0000i		−	1.59237i	−0.605050	−	0.796187i	$-0.706847\pi$
$631$		−	40.0000i		−	1.59237i	0.605050	−	0.796187i	$-0.293153\pi$
$632$	20.0000	+	20.0000i	0.795557	+	0.795557i
$633$		0			0
$634$		−	14.0000i		−	0.556011i
$635$	16.0000	+	16.0000i	0.634941	+	0.634941i
$636$		0			0
$637$		0			0
$638$	−14.0000	+	14.0000i	−0.554265	+	0.554265i
$639$		0			0
$640$		−	32.0000i		−	1.26491i
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$			12.0000i			0.472866i
$645$		0			0
$646$		−	8.00000i		−	0.314756i
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	−18.0000	−	18.0000i	−0.707107	−	0.707107i
$649$			16.0000i			0.628055i
$650$		0			0
$651$		0			0
$652$	−30.0000	−	30.0000i	−1.17489	−	1.17489i
$653$	25.0000	−	25.0000i	0.978326	−	0.978326i	−0.0214444	−	0.999770i	$-0.506827\pi$
$653$	25.0000	−	25.0000i	0.978326	−	0.978326i	0.999770	+	0.0214444i	$0.00682650\pi$
$654$		0			0
$655$	−56.0000			−2.18810
$656$		−	40.0000i		−	1.56174i
$657$	18.0000			0.702247
$658$	−12.0000	−	12.0000i	−0.467809	−	0.467809i
$659$	−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
$661$	26.0000	+	26.0000i	1.01128	+	1.01128i	0.999936	+	0.0113472i	$0.00361200\pi$
$661$	26.0000	+	26.0000i	1.01128	+	1.01128i	0.0113472	+	0.999936i	$0.496388\pi$
$662$	−42.0000			−1.63238
$663$		0			0
$664$	−40.0000			−1.55230
$665$		−	8.00000i		−	0.310227i
$666$			30.0000i			1.16248i
$667$	−42.0000	−	42.0000i	−1.62625	−	1.62625i
$668$	24.0000			0.928588
$669$		0			0
$670$	−12.0000	+	12.0000i	−0.463600	+	0.463600i
$671$	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	$0.272541\pi$
$674$	−8.00000	+	8.00000i	−0.308148	+	0.308148i
$675$		0			0
$676$	26.0000			1.00000
$677$	−20.0000	−	20.0000i	−0.768662	−	0.768662i	0.209209	−	0.977871i	$-0.432911\pi$
$677$	−20.0000	−	20.0000i	−0.768662	−	0.768662i	−0.977871	+	0.209209i	$0.932911\pi$
$678$		0			0
$679$			2.00000i			0.0767530i
$680$		−	16.0000i		−	0.613572i
$681$		0			0
$682$	16.0000			0.612672
$683$	−11.0000	−	11.0000i	−0.420903	−	0.420903i	0.464611	−	0.885515i	$-0.346194\pi$
$683$	−11.0000	−	11.0000i	−0.420903	−	0.420903i	−0.885515	+	0.464611i	$0.846194\pi$
$684$	12.0000	−	12.0000i	0.458831	−	0.458831i
$685$	−24.0000	+	24.0000i	−0.916993	+	0.916993i
$686$	−1.00000	−	1.00000i	−0.0381802	−	0.0381802i
$687$		0			0
$688$	4.00000	+	4.00000i	0.152499	+	0.152499i
$689$		0			0
$690$		0			0
$691$	26.0000	−	26.0000i	0.989087	−	0.989087i	−0.0108545	−	0.999941i	$-0.503455\pi$
$691$	26.0000	−	26.0000i	0.989087	−	0.989087i	0.999941	+	0.0108545i	$0.00345515\pi$
$692$		0			0
$693$	3.00000	+	3.00000i	0.113961	+	0.113961i
$694$	−30.0000			−1.13878
$695$		−	8.00000i		−	0.303457i
$696$		0			0
$697$		−	20.0000i		−	0.757554i
$698$			24.0000i			0.908413i
$699$		0			0
$700$		−	6.00000i		−	0.226779i
$701$	−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
$703$	−20.0000			−0.754314
$704$	−8.00000	+	8.00000i	−0.301511	+	0.301511i
$705$		0			0
$706$	6.00000	−	6.00000i	0.225813	−	0.225813i
$707$	6.00000	−	6.00000i	0.225653	−	0.225653i
$708$		0			0
$709$	−7.00000	−	7.00000i	−0.262891	−	0.262891i	0.563337	−	0.826227i	$-0.309517\pi$
$709$	−7.00000	−	7.00000i	−0.262891	−	0.262891i	−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$			30.0000i			1.12509i
$712$	28.0000	−	28.0000i	1.04934	−	1.04934i
$713$			48.0000i			1.79761i
$714$		0			0
$715$		0			0
$716$	6.00000	+	6.00000i	0.224231	+	0.224231i
$717$		0			0
$718$	−6.00000	−	6.00000i	−0.223918	−	0.223918i
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$	24.0000	−	24.0000i	0.894427	−	0.894427i
$721$	4.00000			0.148968
$722$	−11.0000	−	11.0000i	−0.409378	−	0.409378i
$723$		0			0
$724$	−8.00000	+	8.00000i	−0.297318	+	0.297318i
$725$	21.0000	+	21.0000i	0.779920	+	0.779920i
$726$		0			0
$727$			32.0000i			1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$			32.0000i			1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$			24.0000i			0.888280i
$731$	2.00000	+	2.00000i	0.0739727	+	0.0739727i
$732$		0			0
$733$	−10.0000	+	10.0000i	−0.369358	+	0.369358i	−0.867243	−	0.497885i	$-0.834110\pi$
$733$	−10.0000	+	10.0000i	−0.369358	+	0.369358i	0.497885	+	0.867243i	$0.334110\pi$
$734$	32.0000	−	32.0000i	1.18114	−	1.18114i
$735$		0			0
$736$	−24.0000	−	24.0000i	−0.884652	−	0.884652i
$737$	6.00000			0.221013
$738$	30.0000	−	30.0000i	1.10432	−	1.10432i
$739$	7.00000	−	7.00000i	0.257499	−	0.257499i	−0.566537	−	0.824036i	$-0.691717\pi$
$739$	7.00000	−	7.00000i	0.257499	−	0.257499i	0.824036	+	0.566537i	$0.191717\pi$
$740$	−40.0000			−1.47043
$741$		0			0
$742$		−	2.00000i		−	0.0734223i
$743$		−	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
$745$			12.0000i			0.439646i
$746$	22.0000			0.805477
$747$	−30.0000	−	30.0000i	−1.09764	−	1.09764i
$748$	−4.00000	+	4.00000i	−0.146254	+	0.146254i
$749$	5.00000	−	5.00000i	0.182696	−	0.182696i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$	48.0000			1.75038
$753$		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	$0.875972\pi$
$758$	−26.0000			−0.944363
$759$		0			0
$760$	16.0000	+	16.0000i	0.580381	+	0.580381i
$761$		−	30.0000i		−	1.08750i	−0.839248	−	0.543750i	$-0.817004\pi$
$761$		−	30.0000i		−	1.08750i	0.839248	−	0.543750i	$-0.182996\pi$
$762$		0			0
$763$	3.00000	+	3.00000i	0.108607	+	0.108607i
$764$			36.0000i			1.30243i
$765$	12.0000	−	12.0000i	0.433861	−	0.433861i
$766$	−4.00000	+	4.00000i	−0.144526	+	0.144526i
$767$		0			0
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$	−4.00000	+	4.00000i	−0.144150	+	0.144150i
$771$		0			0
$772$			32.0000i			1.15171i
$773$	−6.00000	−	6.00000i	−0.215805	−	0.215805i	0.590923	−	0.806728i	$-0.298764\pi$
$773$	−6.00000	−	6.00000i	−0.215805	−	0.215805i	−0.806728	+	0.590923i	$0.798764\pi$
$774$			6.00000i			0.215666i
$775$		−	24.0000i		−	0.862105i
$776$	−4.00000	−	4.00000i	−0.143592	−	0.143592i
$777$		0			0
$778$	−6.00000			−0.215110
$779$	20.0000	+	20.0000i	0.716574	+	0.716574i
$780$		0			0
$781$		0			0
$782$	−12.0000	−	12.0000i	−0.429119	−	0.429119i
$783$		0			0
$784$	4.00000			0.142857
$785$	−48.0000			−1.71319
$786$		0			0
$787$	18.0000	−	18.0000i	0.641631	−	0.641631i	−0.309326	−	0.950956i	$-0.600103\pi$
$787$	18.0000	−	18.0000i	0.641631	−	0.641631i	0.950956	+	0.309326i	$0.100103\pi$
$788$	10.0000	−	10.0000i	0.356235	−	0.356235i
$789$		0			0
$790$	−40.0000			−1.42314
$791$		−	4.00000i		−	0.142224i
$792$	−12.0000			−0.426401
$793$		0			0
$794$		−	24.0000i		−	0.851728i
$795$		0			0
$796$	−48.0000			−1.70131
$797$	−32.0000	+	32.0000i	−1.13350	+	1.13350i	−0.143907	+	0.989591i	$0.545967\pi$
$797$	−32.0000	+	32.0000i	−1.13350	+	1.13350i	−0.989591	+	0.143907i	$0.954033\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$	12.0000	+	12.0000i	0.424264	+	0.424264i
$801$	42.0000			1.48400
$802$	−12.0000	+	12.0000i	−0.423735	+	0.423735i
$803$	6.00000	−	6.00000i	0.211735	−	0.211735i
$804$		0			0
$805$	−12.0000	−	12.0000i	−0.422944	−	0.422944i
$806$		0			0
$807$		0			0
$808$			24.0000i			0.844317i
$809$			6.00000i			0.210949i	0.994422	+	0.105474i	$0.0336361\pi$
$809$			6.00000i			0.210949i	−0.994422	+	0.105474i	$0.966364\pi$
$810$	36.0000			1.26491
$811$	−24.0000	−	24.0000i	−0.842754	−	0.842754i	0.146462	−	0.989216i	$-0.453211\pi$
$811$	−24.0000	−	24.0000i	−0.842754	−	0.842754i	−0.989216	+	0.146462i	$0.953211\pi$
$812$	−14.0000	+	14.0000i	−0.491304	+	0.491304i
$813$		0			0
$814$	10.0000	+	10.0000i	0.350500	+	0.350500i
$815$	60.0000			2.10171
$816$		0			0
$817$	−4.00000			−0.139942
$818$	14.0000	+	14.0000i	0.489499	+	0.489499i
$819$		0			0
$820$	40.0000	+	40.0000i	1.39686	+	1.39686i
$821$	1.00000	+	1.00000i	0.0349002	+	0.0349002i	0.724342	−	0.689441i	$-0.242144\pi$
$821$	1.00000	+	1.00000i	0.0349002	+	0.0349002i	−0.689441	+	0.724342i	$0.742144\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	−8.00000	+	8.00000i	−0.278693	+	0.278693i
$825$		0			0
$826$			16.0000i			0.556711i
$827$	−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	$0.789233\pi$
$828$		−	36.0000i		−	1.25109i
$829$	2.00000	−	2.00000i	0.0694629	−	0.0694629i	−0.671522	−	0.740985i	$-0.734359\pi$
$829$	2.00000	−	2.00000i	0.0694629	−	0.0694629i	0.740985	+	0.671522i	$0.234359\pi$
$830$	40.0000	−	40.0000i	1.38842	−	1.38842i
$831$		0			0
$832$		0			0
$833$	2.00000			0.0692959
$834$		0			0
$835$	−24.0000	+	24.0000i	−0.830554	+	0.830554i
$836$		−	8.00000i		−	0.276686i
$837$		0			0
$838$			24.0000i			0.829066i
$839$		−	4.00000i		−	0.138095i	−0.997613	−	0.0690477i	$-0.978004\pi$
$839$		−	4.00000i		−	0.138095i	0.997613	−	0.0690477i	$-0.0219961\pi$
$840$		0			0
$841$		−	69.0000i		−	2.37931i
$842$	18.0000			0.620321
$843$		0			0
$844$	18.0000	+	18.0000i	0.619586	+	0.619586i
$845$	−26.0000	+	26.0000i	−0.894427	+	0.894427i
$846$	36.0000	+	36.0000i	1.23771	+	1.23771i
$847$	−9.00000			−0.309244
$848$	4.00000	+	4.00000i	0.137361	+	0.137361i
$849$		0			0
$850$	6.00000	+	6.00000i	0.205798	+	0.205798i
$851$	−30.0000	+	30.0000i	−1.02839	+	1.02839i
$852$		0			0
$853$	24.0000	+	24.0000i	0.821744	+	0.821744i	0.986358	−	0.164614i	$-0.0526378\pi$
$853$	24.0000	+	24.0000i	0.821744	+	0.821744i	−0.164614	+	0.986358i	$0.552638\pi$
$854$	12.0000			0.410632
$855$			24.0000i			0.820783i
$856$			20.0000i			0.683586i
$857$		−	18.0000i		−	0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$		−	18.0000i		−	0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$		0			0
$859$	−2.00000	−	2.00000i	−0.0682391	−	0.0682391i	0.672164	−	0.740403i	$-0.265365\pi$
$859$	−2.00000	−	2.00000i	−0.0682391	−	0.0682391i	−0.740403	+	0.672164i	$0.765365\pi$
$860$	−8.00000			−0.272798
$861$		0			0
$862$	8.00000	−	8.00000i	0.272481	−	0.272481i
$863$	14.0000			0.476566			0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000			0.476566			0.238283	+	0.971196i	$0.423415\pi$
$864$		0			0
$865$		0			0
$866$	6.00000	−	6.00000i	0.203888	−	0.203888i
$867$		0			0
$868$	16.0000			0.543075
$869$	10.0000	+	10.0000i	0.339227	+	0.339227i
$870$		0			0
$871$		0			0
$872$	−12.0000			−0.406371
$873$		−	6.00000i		−	0.203069i
$874$	24.0000			0.811812
$875$	−4.00000	−	4.00000i	−0.135225	−	0.135225i
$876$		0			0
$877$	3.00000	−	3.00000i	0.101303	−	0.101303i	−0.654639	−	0.755942i	$-0.727179\pi$
$877$	3.00000	−	3.00000i	0.101303	−	0.101303i	0.755942	+	0.654639i	$0.227179\pi$
$878$	−16.0000	−	16.0000i	−0.539974	−	0.539974i
$879$		0			0
$880$		−	16.0000i		−	0.539360i
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$	3.00000	+	3.00000i	0.101015	+	0.101015i
$883$	−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
$886$	2.00000			0.0671913
$887$			12.0000i			0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$			12.0000i			0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$		0			0
$889$		−	8.00000i		−	0.268311i
$890$			56.0000i			1.87712i
$891$	−9.00000	−	9.00000i	−0.301511	−	0.301511i
$892$		−	8.00000i		−	0.267860i
$893$	−24.0000	+	24.0000i	−0.803129	+	0.803129i
$894$		0			0
$895$	−12.0000			−0.401116
$896$	−8.00000	+	8.00000i	−0.267261	+	0.267261i
$897$		0			0
$898$	30.0000	−	30.0000i	1.00111	−	1.00111i
$899$	−56.0000	+	56.0000i	−1.86770	+	1.86770i
$900$			18.0000i			0.600000i
$901$	2.00000	+	2.00000i	0.0666297	+	0.0666297i
$902$		−	20.0000i		−	0.665927i
$903$		0			0
$904$	8.00000	+	8.00000i	0.266076	+	0.266076i
$905$		−	16.0000i		−	0.531858i
$906$		0			0
$907$	15.0000	+	15.0000i	0.498067	+	0.498067i	0.910836	−	0.412769i	$-0.135438\pi$
$907$	15.0000	+	15.0000i	0.498067	+	0.498067i	−0.412769	+	0.910836i	$0.635438\pi$
$908$	4.00000	+	4.00000i	0.132745	+	0.132745i
$909$	−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
$913$	−20.0000			−0.661903
$914$	−22.0000	−	22.0000i	−0.727695	−	0.727695i
$915$		0			0
$916$	16.0000	−	16.0000i	0.528655	−	0.528655i
$917$	14.0000	+	14.0000i	0.462321	+	0.462321i
$918$		0			0
$919$		−	24.0000i		−	0.791687i	−0.918318	−	0.395843i	$-0.870452\pi$
$919$		−	24.0000i		−	0.791687i	0.918318	−	0.395843i	$-0.129548\pi$
$920$	48.0000			1.58251
$921$		0			0
$922$			32.0000i			1.05386i
$923$		0			0
$924$		0			0
$925$	15.0000	−	15.0000i	0.493197	−	0.493197i
$926$	−14.0000	+	14.0000i	−0.460069	+	0.460069i
$927$	−12.0000			−0.394132
$928$		−	56.0000i		−	1.83829i
$929$	−50.0000			−1.64045			−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000			−1.64045			−0.820223	+	0.572043i	$0.806151\pi$
$930$		0			0
$931$	−2.00000	+	2.00000i	−0.0655474	+	0.0655474i
$932$	−32.0000			−1.04819
$933$		0			0
$934$			36.0000i			1.17796i
$935$		−	8.00000i		−	0.261628i
$936$		0			0
$937$		−	58.0000i		−	1.89478i	−0.320085	−	0.947389i	$-0.603712\pi$
$937$		−	58.0000i		−	1.89478i	0.320085	−	0.947389i	$-0.396288\pi$
$938$	6.00000			0.195907
$939$		0			0
$940$	−48.0000	+	48.0000i	−1.56559	+	1.56559i
$941$	26.0000	−	26.0000i	0.847576	−	0.847576i	−0.142254	−	0.989830i	$-0.545435\pi$
$941$	26.0000	−	26.0000i	0.847576	−	0.847576i	0.989830	+	0.142254i	$0.0454351\pi$
$942$		0			0
$943$	60.0000			1.95387
$944$	−32.0000	−	32.0000i	−1.04151	−	1.04151i
$945$		0			0
$946$	2.00000	+	2.00000i	0.0650256	+	0.0650256i
$947$	33.0000	−	33.0000i	1.07236	−	1.07236i	0.0751864	−	0.997169i	$-0.476045\pi$
$947$	33.0000	−	33.0000i	1.07236	−	1.07236i	0.997169	−	0.0751864i	$-0.0239552\pi$
$948$		0			0
$949$		0			0
$950$	−12.0000			−0.389331
$951$		0			0
$952$	−4.00000	+	4.00000i	−0.129641	+	0.129641i
$953$		−	16.0000i		−	0.518291i	−0.965838	−	0.259145i	$-0.916559\pi$
$953$		−	16.0000i		−	0.518291i	0.965838	−	0.259145i	$-0.0834409\pi$
$954$			6.00000i			0.194257i
$955$	−36.0000	−	36.0000i	−1.16493	−	1.16493i
$956$		0			0
$957$		0			0
$958$	20.0000	−	20.0000i	0.646171	−	0.646171i
$959$	12.0000			0.387500
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	−15.0000	+	15.0000i	−0.483368	+	0.483368i
$964$		−	44.0000i		−	1.41714i
$965$	−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	$0.989762\pi$
$968$	18.0000	−	18.0000i	0.578542	−	0.578542i
$969$		0			0
$970$	8.00000			0.256865
$971$	−4.00000	−	4.00000i	−0.128366	−	0.128366i	0.640005	−	0.768371i	$-0.278932\pi$
$971$	−4.00000	−	4.00000i	−0.128366	−	0.128366i	−0.768371	+	0.640005i	$0.778932\pi$
$972$		0			0
$973$	−2.00000	+	2.00000i	−0.0641171	+	0.0641171i
$974$	−22.0000	−	22.0000i	−0.704925	−	0.704925i
$975$		0			0
$976$	−24.0000	+	24.0000i	−0.768221	+	0.768221i
$977$	38.0000			1.21573			0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000			1.21573			0.607864	+	0.794041i	$0.292027\pi$
$978$		0			0
$979$	14.0000	−	14.0000i	0.447442	−	0.447442i
$980$	−4.00000	+	4.00000i	−0.127775	+	0.127775i
$981$	−9.00000	−	9.00000i	−0.287348	−	0.287348i
$982$	38.0000			1.21263
$983$			44.0000i			1.40338i	0.712481	+	0.701691i	$0.247571\pi$
$983$			44.0000i			1.40338i	−0.712481	+	0.701691i	$0.752429\pi$
$984$		0			0
$985$			20.0000i			0.637253i
$986$		−	28.0000i		−	0.891702i
$987$		0			0
$988$		0			0
$989$	−6.00000	+	6.00000i	−0.190789	+	0.190789i
$990$	12.0000	−	12.0000i	0.381385	−	0.381385i
$991$	22.0000			0.698853			0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000			0.698853			0.349427	+	0.936964i	$0.386376\pi$
$992$	−32.0000	+	32.0000i	−1.01600	+	1.01600i
$993$		0			0
$994$		0			0
$995$	48.0000	−	48.0000i	1.52170	−	1.52170i
$996$		0			0
$997$	−30.0000	−	30.0000i	−0.950110	−	0.950110i	0.0487037	−	0.998813i	$-0.484491\pi$
$997$	−30.0000	−	30.0000i	−0.950110	−	0.950110i	−0.998813	+	0.0487037i	$0.984491\pi$
$998$		−	46.0000i		−	1.45610i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.2.m.b.85.1	yes	2
4.3	odd	2		448.2.m.a.113.1		2
7.2	even	3		784.2.x.d.165.1		4
7.3	odd	6		784.2.x.e.373.1		4
7.4	even	3		784.2.x.d.373.1		4
7.5	odd	6		784.2.x.e.165.1		4
7.6	odd	2		784.2.m.a.197.1		2
8.3	odd	2		896.2.m.c.225.1		2
8.5	even	2		896.2.m.b.225.1		2
16.3	odd	4		448.2.m.a.337.1		2
16.5	even	4		896.2.m.b.673.1		2
16.11	odd	4		896.2.m.c.673.1		2
16.13	even	4	inner	112.2.m.b.29.1	✓	2
32.3	odd	8		7168.2.a.k.1.1		2
32.13	even	8		7168.2.a.b.1.2		2
32.19	odd	8		7168.2.a.k.1.2		2
32.29	even	8		7168.2.a.b.1.1		2
112.13	odd	4		784.2.m.a.589.1		2
112.45	odd	12		784.2.x.e.765.1		4
112.61	odd	12		784.2.x.e.557.1		4
112.93	even	12		784.2.x.d.557.1		4
112.109	even	12		784.2.x.d.765.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.b.29.1	✓	2	16.13	even	4	inner
112.2.m.b.85.1	yes	2	1.1	even	1	trivial
448.2.m.a.113.1		2	4.3	odd	2
448.2.m.a.337.1		2	16.3	odd	4
784.2.m.a.197.1		2	7.6	odd	2
784.2.m.a.589.1		2	112.13	odd	4
784.2.x.d.165.1		4	7.2	even	3
784.2.x.d.373.1		4	7.4	even	3
784.2.x.d.557.1		4	112.93	even	12
784.2.x.d.765.1		4	112.109	even	12
784.2.x.e.165.1		4	7.5	odd	6
784.2.x.e.373.1		4	7.3	odd	6
784.2.x.e.557.1		4	112.61	odd	12
784.2.x.e.765.1		4	112.45	odd	12
896.2.m.b.225.1		2	8.5	even	2
896.2.m.b.673.1		2	16.5	even	4
896.2.m.c.225.1		2	8.3	odd	2
896.2.m.c.673.1		2	16.11	odd	4
7168.2.a.b.1.1		2	32.29	even	8
7168.2.a.b.1.2		2	32.13	even	8
7168.2.a.k.1.1		2	32.3	odd	8
7168.2.a.k.1.2		2	32.19	odd	8

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	112.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(2.00000 + 2.00000i) q^{5} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(2.00000 + 2.00000i) q^{5} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} -4.00000 q^{10} +(1.00000 + 1.00000i) q^{11} +(1.00000 + 1.00000i) q^{14} -4.00000 q^{16} -2.00000 q^{17} +(-3.00000 - 3.00000i) q^{18} +(2.00000 - 2.00000i) q^{19} +(4.00000 - 4.00000i) q^{20} -2.00000 q^{22} -6.00000i q^{23} +3.00000i q^{25} -2.00000 q^{28} +(7.00000 - 7.00000i) q^{29} -8.00000 q^{31} +(4.00000 - 4.00000i) q^{32} +(2.00000 - 2.00000i) q^{34} +(2.00000 - 2.00000i) q^{35} +6.00000 q^{36} +(-5.00000 - 5.00000i) q^{37} +4.00000i q^{38} +8.00000i q^{40} +10.0000i q^{41} +(-1.00000 - 1.00000i) q^{43} +(2.00000 - 2.00000i) q^{44} +(-6.00000 + 6.00000i) q^{45} +(6.00000 + 6.00000i) q^{46} -12.0000 q^{47} -1.00000 q^{49} +(-3.00000 - 3.00000i) q^{50} +(-1.00000 - 1.00000i) q^{53} +4.00000i q^{55} +(2.00000 - 2.00000i) q^{56} +14.0000i q^{58} +(8.00000 + 8.00000i) q^{59} +(6.00000 - 6.00000i) q^{61} +(8.00000 - 8.00000i) q^{62} +3.00000 q^{63} +8.00000i q^{64} +(3.00000 - 3.00000i) q^{67} +4.00000i q^{68} +4.00000i q^{70} +(-6.00000 + 6.00000i) q^{72} -6.00000i q^{73} +10.0000 q^{74} +(-4.00000 - 4.00000i) q^{76} +(1.00000 - 1.00000i) q^{77} +10.0000 q^{79} +(-8.00000 - 8.00000i) q^{80} -9.00000 q^{81} +(-10.0000 - 10.0000i) q^{82} +(-10.0000 + 10.0000i) q^{83} +(-4.00000 - 4.00000i) q^{85} +2.00000 q^{86} +4.00000i q^{88} -14.0000i q^{89} -12.0000i q^{90} -12.0000 q^{92} +(12.0000 - 12.0000i) q^{94} +8.00000 q^{95} -2.00000 q^{97} +(1.00000 - 1.00000i) q^{98} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{5} + 4 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 4 * q^5 + 4 * q^8	\( 2 q - 2 q^{2} + 4 q^{5} + 4 q^{8} - 8 q^{10} + 2 q^{11} + 2 q^{14} - 8 q^{16} - 4 q^{17} - 6 q^{18} + 4 q^{19} + 8 q^{20} - 4 q^{22} - 4 q^{28} + 14 q^{29} - 16 q^{31} + 8 q^{32} + 4 q^{34} + 4 q^{35} + 12 q^{36} - 10 q^{37} - 2 q^{43} + 4 q^{44} - 12 q^{45} + 12 q^{46} - 24 q^{47} - 2 q^{49} - 6 q^{50} - 2 q^{53} + 4 q^{56} + 16 q^{59} + 12 q^{61} + 16 q^{62} + 6 q^{63} + 6 q^{67} - 12 q^{72} + 20 q^{74} - 8 q^{76} + 2 q^{77} + 20 q^{79} - 16 q^{80} - 18 q^{81} - 20 q^{82} - 20 q^{83} - 8 q^{85} + 4 q^{86} - 24 q^{92} + 24 q^{94} + 16 q^{95} - 4 q^{97} + 2 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^5 + 4 * q^8 - 8 * q^10 + 2 * q^11 + 2 * q^14 - 8 * q^16 - 4 * q^17 - 6 * q^18 + 4 * q^19 + 8 * q^20 - 4 * q^22 - 4 * q^28 + 14 * q^29 - 16 * q^31 + 8 * q^32 + 4 * q^34 + 4 * q^35 + 12 * q^36 - 10 * q^37 - 2 * q^43 + 4 * q^44 - 12 * q^45 + 12 * q^46 - 24 * q^47 - 2 * q^49 - 6 * q^50 - 2 * q^53 + 4 * q^56 + 16 * q^59 + 12 * q^61 + 16 * q^62 + 6 * q^63 + 6 * q^67 - 12 * q^72 + 20 * q^74 - 8 * q^76 + 2 * q^77 + 20 * q^79 - 16 * q^80 - 18 * q^81 - 20 * q^82 - 20 * q^83 - 8 * q^85 + 4 * q^86 - 24 * q^92 + 24 * q^94 + 16 * q^95 - 4 * q^97 + 2 * q^98 - 6 * q^99

Introduction

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