LMFDB - Embedded newform 799.1.h.b.610.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [799,1,Mod(93,799)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(799, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([5, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("799.93");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 799 = 17 \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	799.h (of order $8$, degree $4$, 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:	$0.398752945094$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} + \cdots)$

Embedding invariants

Embedding label			610.2
Root			$-0.987688 - 0.156434i$ of defining polynomial
Character	$\chi$	$=$	799.610
Dual form			799.1.h.b.93.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.831254 - 0.831254i) q^{2} +(0.763007 + 1.84206i) q^{3} +0.381966i q^{4} +(0.896969 - 2.16547i) q^{6} +(0.431351 + 0.178671i) q^{7} +(-0.513743 + 0.513743i) q^{8} +(-2.10391 + 2.10391i) q^{9} +O(q^{10})$	\(q+(-0.831254 - 0.831254i) q^{2} +(0.763007 + 1.84206i) q^{3} +0.381966i q^{4} +(0.896969 - 2.16547i) q^{6} +(0.431351 + 0.178671i) q^{7} +(-0.513743 + 0.513743i) q^{8} +(-2.10391 + 2.10391i) q^{9} +(-0.703605 + 0.291443i) q^{12} +(-0.210041 - 0.507083i) q^{14} +1.23607 q^{16} +(-0.309017 + 0.951057i) q^{17} +3.49777 q^{18} +0.930903i q^{21} +(-1.33834 - 0.554357i) q^{24} +(0.707107 - 0.707107i) q^{25} +(-3.63877 - 1.50723i) q^{27} +(-0.0682464 + 0.164761i) q^{28} +(-0.513743 - 0.513743i) q^{32} +(1.04744 - 0.533698i) q^{34} +(-0.803622 - 0.803622i) q^{36} +(-0.0600500 - 0.144974i) q^{37} +(0.773817 - 0.773817i) q^{42} +1.00000i q^{47} +(0.943129 + 2.27692i) q^{48} +(-0.552967 - 0.552967i) q^{49} -1.17557 q^{50} +(-1.98769 + 0.156434i) q^{51} +(1.39680 + 1.39680i) q^{53} +(1.77185 + 4.27763i) q^{54} +(-0.313395 + 0.129812i) q^{56} +(1.14412 - 1.14412i) q^{59} +(0.965451 + 0.399903i) q^{61} +(-1.28343 + 0.531615i) q^{63} -0.381966i q^{64} +(-0.363271 - 0.118034i) q^{68} +(-0.581990 - 1.40505i) q^{71} -2.16174i q^{72} +(-0.0705930 + 0.170427i) q^{74} +(1.84206 + 0.763007i) q^{75} +(-0.581990 + 1.40505i) q^{79} -4.87749i q^{81} +(-1.00000 - 1.00000i) q^{83} -0.355573 q^{84} -1.17557i q^{89} +(0.831254 - 0.831254i) q^{94} +(0.554357 - 1.33834i) q^{96} +(1.57547 - 0.652583i) q^{97} +0.919311i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^3 - 4 * q^9	$ 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71} - 4 q^{79} - 16 q^{83} + 32 q^{84}+O(q^{100}) $ 16 * q - 4 * q^3 - 4 * q^9 - 16 * q^16 + 4 * q^17 - 20 * q^24 - 4 * q^27 - 4 * q^28 + 4 * q^36 - 20 * q^42 + 24 * q^48 + 4 * q^49 - 16 * q^51 + 4 * q^53 + 20 * q^54 + 20 * q^56 + 4 * q^61 - 4 * q^63 - 4 * q^71 - 4 * q^79 - 16 * q^83 + 32 * q^84

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/799\mathbb{Z}\right)^\times$.

$n$	$52$	$377$
$\chi(n)$	$-1$	$e\left(\frac{3}{8}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$2$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$3$	0.763007	+	1.84206i	0.763007	+	1.84206i	0.453990	+	0.891007i	$0.350000\pi$
$3$	0.763007	+	1.84206i	0.763007	+	1.84206i	0.309017	+	0.951057i	$0.400000\pi$
$4$			0.381966i			0.381966i
$5$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$5$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$6$	0.896969	−	2.16547i	0.896969	−	2.16547i
$7$	0.431351	+	0.178671i	0.431351	+	0.178671i	0.587785	−	0.809017i	$-0.300000\pi$
$7$	0.431351	+	0.178671i	0.431351	+	0.178671i	−0.156434	+	0.987688i	$0.550000\pi$
$8$	−0.513743	+	0.513743i	−0.513743	+	0.513743i
$9$	−2.10391	+	2.10391i	−2.10391	+	2.10391i
$10$		0			0
$11$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$11$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$12$	−0.703605	+	0.291443i	−0.703605	+	0.291443i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−0.210041	−	0.507083i	−0.210041	−	0.507083i
$15$		0			0
$16$	1.23607			1.23607
$17$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$17$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$18$	3.49777			3.49777
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$			0.930903i			0.930903i
$22$		0			0
$23$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$23$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$24$	−1.33834	−	0.554357i	−1.33834	−	0.554357i
$25$	0.707107	−	0.707107i	0.707107	−	0.707107i
$26$		0			0
$27$	−3.63877	−	1.50723i	−3.63877	−	1.50723i
$28$	−0.0682464	+	0.164761i	−0.0682464	+	0.164761i
$29$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$29$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$30$		0			0
$31$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$31$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$32$	−0.513743	−	0.513743i	−0.513743	−	0.513743i
$33$		0			0
$34$	1.04744	−	0.533698i	1.04744	−	0.533698i
$35$		0			0
$36$	−0.803622	−	0.803622i	−0.803622	−	0.803622i
$37$	−0.0600500	−	0.144974i	−0.0600500	−	0.144974i	0.891007	−	0.453990i	$-0.150000\pi$
$37$	−0.0600500	−	0.144974i	−0.0600500	−	0.144974i	−0.951057	+	0.309017i	$0.900000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$41$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$42$	0.773817	−	0.773817i	0.773817	−	0.773817i
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.00000i			1.00000i
$47$			1.00000i			1.00000i
$48$	0.943129	+	2.27692i	0.943129	+	2.27692i
$49$	−0.552967	−	0.552967i	−0.552967	−	0.552967i
$50$	−1.17557			−1.17557
$51$	−1.98769	+	0.156434i	−1.98769	+	0.156434i
$52$		0			0
$53$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$53$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$54$	1.77185	+	4.27763i	1.77185	+	4.27763i
$55$		0			0
$56$	−0.313395	+	0.129812i	−0.313395	+	0.129812i
$57$		0			0
$58$		0			0
$59$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$59$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$60$		0			0
$61$	0.965451	+	0.399903i	0.965451	+	0.399903i	0.809017	−	0.587785i	$-0.200000\pi$
$61$	0.965451	+	0.399903i	0.965451	+	0.399903i	0.156434	+	0.987688i	$0.450000\pi$
$62$		0			0
$63$	−1.28343	+	0.531615i	−1.28343	+	0.531615i
$64$		−	0.381966i		−	0.381966i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−0.363271	−	0.118034i	−0.363271	−	0.118034i
$69$		0			0
$70$		0			0
$71$	−0.581990	−	1.40505i	−0.581990	−	1.40505i	−0.891007	−	0.453990i	$-0.850000\pi$
$71$	−0.581990	−	1.40505i	−0.581990	−	1.40505i	0.309017	−	0.951057i	$-0.400000\pi$
$72$		−	2.16174i		−	2.16174i
$73$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$73$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$74$	−0.0705930	+	0.170427i	−0.0705930	+	0.170427i
$75$	1.84206	+	0.763007i	1.84206	+	0.763007i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	0.309017	+	0.951057i	$0.400000\pi$
$79$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	−0.891007	+	0.453990i	$0.850000\pi$
$80$		0			0
$81$		−	4.87749i		−	4.87749i
$82$		0			0
$83$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$83$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$84$	−0.355573			−0.355573
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$89$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.831254	−	0.831254i	0.831254	−	0.831254i
$95$		0			0
$96$	0.554357	−	1.33834i	0.554357	−	1.33834i
$97$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.587785	−	0.809017i	$-0.300000\pi$
$97$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.987688	+	0.156434i	$0.0500000\pi$
$98$			0.919311i			0.919311i
$99$		0			0
$100$	0.270091	+	0.270091i	0.270091	+	0.270091i
$101$	0.907981			0.907981			0.453990	−	0.891007i	$-0.350000\pi$
$101$	0.907981			0.907981			0.453990	+	0.891007i	$0.350000\pi$
$102$	1.78231	+	1.52224i	1.78231	+	1.52224i
$103$	1.78201			1.78201			0.891007	−	0.453990i	$-0.150000\pi$
$103$	1.78201			1.78201			0.891007	+	0.453990i	$0.150000\pi$
$104$		0			0
$105$		0			0
$106$		−	2.32219i		−	2.32219i
$107$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$107$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$108$	0.575710	−	1.38989i	0.575710	−	1.38989i
$109$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$109$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$110$		0			0
$111$	0.221232	−	0.221232i	0.221232	−	0.221232i
$112$	0.533179	+	0.220850i	0.533179	+	0.220850i
$113$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$113$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−1.90211			−1.90211
$119$	−0.303221	+	0.355026i	−0.303221	+	0.355026i
$120$		0			0
$121$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$122$	−0.470114	−	1.13496i	−0.470114	−	1.13496i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1.50876	+	0.624950i	1.50876	+	0.624950i
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−0.831254	+	0.831254i	−0.831254	+	0.831254i
$129$		0			0
$130$		0			0
$131$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.987688	−	0.156434i	$-0.950000\pi$
$131$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.587785	+	0.809017i	$0.700000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.329843	−	0.647354i	−0.329843	−	0.647354i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$139$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$140$		0			0
$141$	−1.84206	+	0.763007i	−1.84206	+	0.763007i
$142$	−0.684170	+	1.65173i	−0.684170	+	1.65173i
$143$		0			0
$144$	−2.60057	+	2.60057i	−2.60057	+	2.60057i
$145$		0			0
$146$		0			0
$147$	0.596682	−	1.44052i	0.596682	−	1.44052i
$148$	0.0553750	−	0.0229371i	0.0553750	−	0.0229371i
$149$		−	1.78201i		−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$149$		−	1.78201i		−	1.78201i	0.453990	−	0.891007i	$-0.350000\pi$
$150$	−0.896969	−	2.16547i	−0.896969	−	2.16547i
$151$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$151$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$152$		0			0
$153$	−1.35079	−	2.65108i	−1.35079	−	2.65108i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	0.312869i		−	0.312869i	−0.987688	−	0.156434i	$-0.950000\pi$
$157$		−	0.312869i		−	0.312869i	0.987688	−	0.156434i	$-0.0500000\pi$
$158$	1.65173	−	0.684170i	1.65173	−	0.684170i
$159$	−1.50723	+	3.63877i	−1.50723	+	3.63877i
$160$		0			0
$161$		0			0
$162$	−4.05443	+	4.05443i	−4.05443	+	4.05443i
$163$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$163$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$164$		0			0
$165$		0			0
$166$			1.66251i			1.66251i
$167$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$167$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$168$	−0.478245	−	0.478245i	−0.478245	−	0.478245i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.744220	−	1.79671i	−0.744220	−	1.79671i	−0.587785	−	0.809017i	$-0.700000\pi$
$173$	−0.744220	−	1.79671i	−0.744220	−	1.79671i	−0.156434	−	0.987688i	$-0.550000\pi$
$174$		0			0
$175$	0.431351	−	0.178671i	0.431351	−	0.178671i
$176$		0			0
$177$	2.98052	+	1.23457i	2.98052	+	1.23457i
$178$	−0.977198	+	0.977198i	−0.977198	+	0.977198i
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$181$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$182$		0			0
$183$			2.08355i			2.08355i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.381966			−0.381966
$189$	−1.30029	−	1.30029i	−1.30029	−	1.30029i
$190$		0			0
$191$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$191$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$192$	0.703605	−	0.291443i	0.703605	−	0.291443i
$193$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$193$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$194$	−1.85208	−	0.767157i	−1.85208	−	0.767157i
$195$		0			0
$196$	0.211214	−	0.211214i	0.211214	−	0.211214i
$197$	1.70711	+	0.707107i	1.70711	+	0.707107i	1.00000			$0$
$197$	1.70711	+	0.707107i	1.70711	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$199$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$200$			0.726543i			0.726543i
$201$		0			0
$202$	−0.754763	−	0.754763i	−0.754763	−	0.754763i
$203$		0			0
$204$	−0.0597526	−	0.759229i	−0.0597526	−	0.759229i
$205$		0			0
$206$	−1.48131	−	1.48131i	−1.48131	−	1.48131i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$211$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$212$	−0.533531	+	0.533531i	−0.533531	+	0.533531i
$213$	2.14412	−	2.14412i	2.14412	−	2.14412i
$214$		0			0
$215$		0			0
$216$	2.64372	−	1.09506i	2.64372	−	1.09506i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	−0.367799			−0.367799
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$	−0.129812	−	0.313395i	−0.129812	−	0.313395i
$225$			2.97538i			2.97538i
$226$		0			0
$227$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$227$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$228$		0			0
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$233$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$234$		0			0
$235$		0			0
$236$	0.437016	+	0.437016i	0.437016	+	0.437016i
$237$	−3.03225			−3.03225
$238$	0.547171	−	0.0430633i	0.547171	−	0.0430633i
$239$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$239$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$240$		0			0
$241$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	0.951057	−	0.309017i	$-0.100000\pi$
$241$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	−0.891007	+	0.453990i	$0.850000\pi$
$242$			1.17557i			1.17557i
$243$	5.34588	−	2.21433i	5.34588	−	2.21433i
$244$	−0.152749	+	0.368770i	−0.152749	+	0.368770i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	1.07906	−	2.60507i	1.07906	−	2.60507i
$250$		0			0
$251$		−	0.907981i		−	0.907981i	−0.891007	−	0.453990i	$-0.850000\pi$
$251$		−	0.907981i		−	0.907981i	0.891007	−	0.453990i	$-0.150000\pi$
$252$	−0.203059	−	0.490227i	−0.203059	−	0.490227i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		−	0.0732636i		−	0.0732636i
$260$		0			0
$261$		0			0
$262$	1.85208	+	0.767157i	1.85208	+	0.767157i
$263$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$263$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.16547	−	0.896969i	2.16547	−	0.896969i
$268$		0			0
$269$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$269$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$270$		0			0
$271$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$271$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$272$	−0.381966	+	1.17557i	−0.381966	+	1.17557i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.951057	−	0.309017i	$-0.900000\pi$
$277$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.453990	+	0.891007i	$0.650000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$	2.16547	+	0.896969i	2.16547	+	0.896969i
$283$	0.178671	−	0.431351i	0.178671	−	0.431351i	−0.809017	−	0.587785i	$-0.800000\pi$
$283$	0.178671	−	0.431351i	0.178671	−	0.431351i	0.987688	+	0.156434i	$0.0500000\pi$
$284$	0.536680	−	0.222300i	0.536680	−	0.222300i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.16174			2.16174
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$		0			0
$291$	2.40420	+	2.40420i	2.40420	+	2.40420i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−1.69343	+	0.701442i	−1.69343	+	0.701442i
$295$		0			0
$296$	0.105329	+	0.0436289i	0.105329	+	0.0436289i
$297$		0			0
$298$	−1.48131	+	1.48131i	−1.48131	+	1.48131i
$299$		0			0
$300$	−0.291443	+	0.703605i	−0.291443	+	0.703605i
$301$		0			0
$302$		0			0
$303$	0.692796	+	1.67256i	0.692796	+	1.67256i
$304$		0			0
$305$		0			0
$306$	−1.08087	+	3.32657i	−1.08087	+	3.32657i
$307$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$307$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$308$		0			0
$309$	1.35969	+	3.28258i	1.35969	+	3.28258i
$310$		0			0
$311$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$311$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$312$		0			0
$313$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$313$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$314$	−0.260074	+	0.260074i	−0.260074	+	0.260074i
$315$		0			0
$316$	−0.536680	−	0.222300i	−0.536680	−	0.222300i
$317$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$317$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$318$	4.27763	−	1.77185i	4.27763	−	1.77185i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.86304			1.86304
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.178671	+	0.431351i	−0.178671	+	0.431351i
$330$		0			0
$331$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$331$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$332$	0.381966	−	0.381966i	0.381966	−	0.381966i
$333$	0.431351	+	0.178671i	0.431351	+	0.178671i
$334$		0			0
$335$		0			0
$336$			1.15066i			1.15066i
$337$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.587785	+	0.809017i	$0.300000\pi$
$337$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.156434	+	0.987688i	$0.450000\pi$
$338$	0.831254	+	0.831254i	0.831254	+	0.831254i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.318395	−	0.768673i	−0.318395	−	0.768673i
$344$		0			0
$345$		0			0
$346$	−0.874883	+	2.11215i	−0.874883	+	2.11215i
$347$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.951057	−	0.309017i	$-0.900000\pi$
$347$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.891007	−	0.453990i	$-0.850000\pi$
$348$		0			0
$349$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$349$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$350$	−0.507083	−	0.210041i	−0.507083	−	0.210041i
$351$		0			0
$352$		0			0
$353$			0.312869i			0.312869i	0.987688	+	0.156434i	$0.0500000\pi$
$353$			0.312869i			0.312869i	−0.987688	+	0.156434i	$0.950000\pi$
$354$	−1.45133	−	3.50381i	−1.45133	−	3.50381i
$355$		0			0
$356$	0.449028			0.449028
$357$	−0.885341	−	0.287665i	−0.885341	−	0.287665i
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$		0			0
$361$			1.00000i			1.00000i
$362$		0			0
$363$	0.763007	−	1.84206i	0.763007	−	1.84206i
$364$		0			0
$365$		0			0
$366$	1.73196	−	1.73196i	1.73196	−	1.73196i
$367$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$367$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.352943	+	0.852080i	0.352943	+	0.852080i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−0.513743	−	0.513743i	−0.513743	−	0.513743i
$377$		0			0
$378$			2.16174i			2.16174i
$379$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.809017	+	0.587785i	$0.800000\pi$
$379$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.987688	+	0.156434i	$0.950000\pi$
$380$		0			0
$381$		0			0
$382$	1.17557	−	1.17557i	1.17557	−	1.17557i
$383$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.309017	−	0.951057i	$-0.400000\pi$
$383$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.951057	−	0.309017i	$-0.100000\pi$
$384$	−2.16547	−	0.896969i	−2.16547	−	0.896969i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.249264	+	0.601777i	0.249264	+	0.601777i
$389$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$389$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		0			0
$392$	0.568166			0.568166
$393$	−2.40420	−	2.40420i	−2.40420	−	2.40420i
$394$	−0.831254	−	2.00682i	−0.831254	−	2.00682i
$395$		0			0
$396$		0			0
$397$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	0.453990	+	0.891007i	$0.350000\pi$
$397$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	−0.951057	+	0.309017i	$0.900000\pi$
$398$		0			0
$399$		0			0
$400$	0.874032	−	0.874032i	0.874032	−	0.874032i
$401$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.951057	+	0.309017i	$0.100000\pi$
$401$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.891007	+	0.453990i	$0.150000\pi$
$402$		0			0
$403$		0			0
$404$			0.346818i			0.346818i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	0.940794	−	1.10153i	0.940794	−	1.10153i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$			0.680668i			0.680668i
$413$	0.697940	−	0.289096i	0.697940	−	0.289096i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$419$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−2.10391	−	2.10391i	−2.10391	−	2.10391i
$424$	−1.43520			−1.43520
$425$	0.453990	+	0.891007i	0.453990	+	0.891007i
$426$	−3.56462			−3.56462
$427$	0.344997	+	0.344997i	0.344997	+	0.344997i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	0.156434	+	0.987688i	$0.450000\pi$
$431$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	−0.809017	+	0.587785i	$0.800000\pi$
$432$	−4.49777	−	1.86304i	−4.49777	−	1.86304i
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$439$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$440$		0			0
$441$	2.32678			2.32678
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	0.0845030	+	0.0845030i	0.0845030	+	0.0845030i
$445$		0			0
$446$		0			0
$447$	3.28258	−	1.35969i	3.28258	−	1.35969i
$448$	0.0682464	−	0.164761i	0.0682464	−	0.164761i
$449$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$449$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$450$	2.47329	−	2.47329i	2.47329	−	2.47329i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$457$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$458$		0			0
$459$	2.55790	−	2.99492i	2.55790	−	2.99492i
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.576324	−	0.238721i	0.576324	−	0.238721i
$472$			1.17557i			1.17557i
$473$		0			0
$474$	2.52057	+	2.52057i	2.52057	+	2.52057i
$475$		0			0
$476$	−0.135608	−	0.115820i	−0.135608	−	0.115820i
$477$	−5.87749			−5.87749
$478$	−1.58114	−	1.58114i	−1.58114	−	1.58114i
$479$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	0.809017	−	0.587785i	$-0.200000\pi$
$479$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	−0.987688	+	0.156434i	$0.950000\pi$
$480$		0			0
$481$		0			0
$482$	0.0705930	−	0.170427i	0.0705930	−	0.170427i
$483$		0			0
$484$	0.270091	−	0.270091i	0.270091	−	0.270091i
$485$		0			0
$486$	−6.28445	−	2.60311i	−6.28445	−	2.60311i
$487$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$487$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$488$	−0.701442	+	0.290547i	−0.701442	+	0.290547i
$489$		0			0
$490$		0			0
$491$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$491$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	0.710053i		−	0.710053i
$498$	−3.06244	+	1.26851i	−3.06244	+	1.26851i
$499$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$499$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$500$		0			0
$501$		0			0
$502$	−0.754763	+	0.754763i	−0.754763	+	0.754763i
$503$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$503$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$504$	0.386241	−	0.932467i	0.386241	−	0.932467i
$505$		0			0
$506$		0			0
$507$	−0.763007	−	1.84206i	−0.763007	−	1.84206i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−0.0609007	+	0.0609007i	−0.0609007	+	0.0609007i
$519$	2.74180	−	2.74180i	2.74180	−	2.74180i
$520$		0			0
$521$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.156434	+	0.987688i	$0.550000\pi$
$521$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.587785	+	0.809017i	$0.700000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−0.249264	−	0.601777i	−0.249264	−	0.601777i
$525$	0.658248	+	0.658248i	0.658248	+	0.658248i
$526$	1.90211			1.90211
$527$		0			0
$528$		0			0
$529$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$530$		0			0
$531$			4.81426i			4.81426i
$532$		0			0
$533$		0			0
$534$	−2.54567	−	1.05445i	−2.54567	−	1.05445i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.344317	+	0.831254i	−0.344317	+	0.831254i
$539$		0			0
$540$		0			0
$541$	−0.744220	−	1.79671i	−0.744220	−	1.79671i	−0.587785	−	0.809017i	$-0.700000\pi$
$541$	−0.744220	−	1.79671i	−0.744220	−	1.79671i	−0.156434	−	0.987688i	$-0.550000\pi$
$542$	0.754763	+	0.754763i	0.754763	+	0.754763i
$543$		0			0
$544$	0.647354	−	0.329843i	0.647354	−	0.329843i
$545$		0			0
$546$		0			0
$547$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$547$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$548$		0			0
$549$	−2.87258	+	1.18986i	−2.87258	+	1.18986i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.502083	+	0.502083i	−0.502083	+	0.502083i
$554$	1.65173	+	0.684170i	1.65173	+	0.684170i
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$	−0.291443	−	0.703605i	−0.291443	−	0.703605i
$565$		0			0
$566$	−0.507083	+	0.210041i	−0.507083	+	0.210041i
$567$	0.871468	−	2.10391i	0.871468	−	2.10391i
$568$	1.02083	+	0.422840i	1.02083	+	0.422840i
$569$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$569$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$570$		0			0
$571$	−1.20002	−	0.497066i	−1.20002	−	0.497066i	−0.309017	−	0.951057i	$-0.600000\pi$
$571$	−1.20002	−	0.497066i	−1.20002	−	0.497066i	−0.891007	+	0.453990i	$0.850000\pi$
$572$		0			0
$573$	−2.60507	+	1.07906i	−2.60507	+	1.07906i
$574$		0			0
$575$		0			0
$576$	0.803622	+	0.803622i	0.803622	+	0.803622i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.183900	+	1.16110i	0.183900	+	1.16110i
$579$		0			0
$580$		0			0
$581$	−0.252679	−	0.610022i	−0.252679	−	0.610022i
$582$		−	3.99700i		−	3.99700i
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	0.550229	+	0.227912i	0.550229	+	0.227912i
$589$		0			0
$590$		0			0
$591$			3.68413i			3.68413i
$592$	−0.0742259	−	0.179197i	−0.0742259	−	0.179197i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$	0.680668			0.680668
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−1.33834	+	0.554357i	−1.33834	+	0.554357i
$601$	0.399903	−	0.965451i	0.399903	−	0.965451i	−0.587785	−	0.809017i	$-0.700000\pi$
$601$	0.399903	−	0.965451i	0.399903	−	0.965451i	0.987688	−	0.156434i	$-0.0500000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	0.814431	−	1.96621i	0.814431	−	1.96621i
$607$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$607$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1.01262	−	0.515957i	1.01262	−	0.515957i
$613$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$613$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$614$	0.754763	+	0.754763i	0.754763	+	0.754763i
$615$		0			0
$616$		0			0
$617$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.309017	−	0.951057i	$-0.400000\pi$
$617$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.891007	+	0.453990i	$0.150000\pi$
$618$	1.59841	−	3.85890i	1.59841	−	3.85890i
$619$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$619$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.210041	−	0.507083i	0.210041	−	0.507083i
$624$		0			0
$625$		−	1.00000i		−	1.00000i
$626$		0			0
$627$		0			0
$628$	0.119505			0.119505
$629$	0.156434	−	0.0123117i	0.156434	−	0.0123117i
$630$		0			0
$631$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$631$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$632$	−0.422840	−	1.02083i	−0.422840	−	1.02083i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−1.38989	−	0.575710i	−1.38989	−	0.575710i
$637$		0			0
$638$		0			0
$639$	4.18054	+	1.73164i	4.18054	+	1.73164i
$640$		0			0
$641$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$641$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$642$		0			0
$643$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	−0.987688	−	0.156434i	$-0.950000\pi$
$643$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	0.587785	−	0.809017i	$-0.300000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.312869			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$647$	0.312869			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$648$	2.50578	+	2.50578i	2.50578	+	2.50578i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.144974	+	0.0600500i	0.144974	+	0.0600500i	0.453990	−	0.891007i	$-0.350000\pi$
$653$	0.144974	+	0.0600500i	0.144974	+	0.0600500i	−0.309017	+	0.951057i	$0.600000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.507083	−	0.210041i	0.507083	−	0.210041i
$659$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$659$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$660$		0			0
$661$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$661$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$662$	1.38197			1.38197
$663$		0			0
$664$	1.02749			1.02749
$665$		0			0
$666$	−0.210041	−	0.507083i	−0.210041	−	0.507083i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	0.478245	−	0.478245i	0.478245	−	0.478245i
$673$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$673$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$674$	0.874883	−	2.11215i	0.874883	−	2.11215i
$675$	−3.63877	+	1.50723i	−3.63877	+	1.50723i
$676$		−	0.381966i		−	0.381966i
$677$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$677$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$678$		0			0
$679$	0.796180			0.796180
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.399903	+	0.965451i	0.399903	+	0.965451i	0.987688	+	0.156434i	$0.0500000\pi$
$683$	0.399903	+	0.965451i	0.399903	+	0.965451i	−0.587785	+	0.809017i	$0.700000\pi$
$684$		0			0
$685$		0			0
$686$	−0.374295	+	0.903629i	−0.374295	+	0.903629i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$691$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$692$	0.686280	−	0.284267i	0.686280	−	0.284267i
$693$		0			0
$694$	0.896969	+	2.16547i	0.896969	+	2.16547i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.0682464	+	0.164761i	0.0682464	+	0.164761i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.260074	−	0.260074i	0.260074	−	0.260074i
$707$	0.391658	+	0.162230i	0.391658	+	0.162230i
$708$	−0.471565	+	1.13846i	−0.471565	+	1.13846i
$709$	0.431351	−	0.178671i	0.431351	−	0.178671i	−0.156434	−	0.987688i	$-0.550000\pi$
$709$	0.431351	−	0.178671i	0.431351	−	0.178671i	0.587785	+	0.809017i	$0.300000\pi$
$710$		0			0
$711$	−1.73164	−	4.18054i	−1.73164	−	4.18054i
$712$	0.603941	+	0.603941i	0.603941	+	0.603941i
$713$		0			0
$714$	0.496821	+	0.975066i	0.496821	+	0.975066i
$715$		0			0
$716$		0			0
$717$	1.45133	+	3.50381i	1.45133	+	3.50381i
$718$		0			0
$719$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.809017	−	0.587785i	$-0.800000\pi$
$719$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.156434	+	0.987688i	$0.550000\pi$
$720$		0			0
$721$	0.768673	+	0.318395i	0.768673	+	0.318395i
$722$	0.831254	−	0.831254i	0.831254	−	0.831254i
$723$	−0.221232	+	0.221232i	−0.221232	+	0.221232i
$724$		0			0
$725$		0			0
$726$	−2.16547	+	0.896969i	−2.16547	+	0.896969i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	4.70898	+	4.70898i	4.70898	+	4.70898i
$730$		0			0
$731$		0			0
$732$	−0.795846			−0.795846
$733$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$733$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.221232	−	0.221232i	0.221232	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$739$	0.221232	−	0.221232i	0.221232	−	0.221232i	0.809017	+	0.587785i	$0.200000\pi$
$740$		0			0
$741$		0			0
$742$	0.414910	−	1.00168i	0.414910	−	1.00168i
$743$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$743$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.20782			4.20782
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$751$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$752$			1.23607i			1.23607i
$753$	1.67256	−	0.692796i	1.67256	−	0.692796i
$754$		0			0
$755$		0			0
$756$	0.496666	−	0.496666i	0.496666	−	0.496666i
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	2.11215	+	0.874883i	2.11215	+	0.874883i
$759$		0			0
$760$		0			0
$761$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$761$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$762$		0			0
$763$		0			0
$764$	−0.540182			−0.540182
$765$		0			0
$766$	−2.09488			−2.09488
$767$		0			0
$768$	0.763007	+	1.84206i	0.763007	+	1.84206i
$769$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$769$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$773$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$774$		0			0
$775$		0			0
$776$	−0.474129	+	1.14465i	−0.474129	+	1.14465i
$777$	0.134956	−	0.0559007i	0.134956	−	0.0559007i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.683504	−	0.683504i	−0.683504	−	0.683504i
$785$		0			0
$786$			3.99700i			3.99700i
$787$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$787$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$788$	−0.270091	+	0.652057i	−0.270091	+	0.652057i
$789$	−2.98052	−	1.23457i	−2.98052	−	1.23457i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	1.41071	−	0.584336i	1.41071	−	0.584336i
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$800$	−0.726543			−0.726543
$801$	2.47329	+	2.47329i	2.47329	+	2.47329i
$802$	−0.896969	−	2.16547i	−0.896969	−	2.16547i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.07906	−	1.07906i	1.07906	−	1.07906i
$808$	−0.466469	+	0.466469i	−0.466469	+	0.466469i
$809$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$809$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$810$		0			0
$811$	1.40505	−	0.581990i	1.40505	−	0.581990i	0.453990	−	0.891007i	$-0.350000\pi$
$811$	1.40505	−	0.581990i	1.40505	−	0.581990i	0.951057	+	0.309017i	$0.100000\pi$
$812$		0			0
$813$	−0.692796	−	1.67256i	−0.692796	−	1.67256i
$814$		0			0
$815$		0			0
$816$	−2.45692	+	0.193364i	−2.45692	+	0.193364i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$821$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$822$		0			0
$823$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$823$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$824$	−0.915497	+	0.915497i	−0.915497	+	0.915497i
$825$		0			0
$826$	−0.820478	−	0.339853i	−0.820478	−	0.339853i
$827$	0.399903	−	0.965451i	0.399903	−	0.965451i	−0.587785	−	0.809017i	$-0.700000\pi$
$827$	0.399903	−	0.965451i	0.399903	−	0.965451i	0.987688	−	0.156434i	$-0.0500000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−2.14412	−	2.14412i	−2.14412	−	2.14412i
$832$		0			0
$833$	0.696779	−	0.355026i	0.696779	−	0.355026i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$839$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$840$		0			0
$841$	0.707107	−	0.707107i	0.707107	−	0.707107i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$			3.49777i			3.49777i
$847$	−0.178671	−	0.431351i	−0.178671	−	0.431351i
$848$	1.72654	+	1.72654i	1.72654	+	1.72654i
$849$	0.930903			0.930903
$850$	0.363271	−	1.11803i	0.363271	−	1.11803i
$851$		0			0
$852$	0.818982	+	0.818982i	0.818982	+	0.818982i
$853$	0.497066	+	1.20002i	0.497066	+	1.20002i	0.951057	+	0.309017i	$0.100000\pi$
$853$	0.497066	+	1.20002i	0.497066	+	1.20002i	−0.453990	+	0.891007i	$0.650000\pi$
$854$		−	0.573560i		−	0.573560i
$855$		0			0
$856$		0			0
$857$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$857$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$	1.85208	−	0.767157i	1.85208	−	0.767157i
$863$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$863$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$864$	1.09506	+	2.64372i	1.09506	+	2.64372i
$865$		0			0
$866$		0			0
$867$	0.465451	−	1.93874i	0.465451	−	1.93874i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.94168	+	4.68763i	−1.94168	+	4.68763i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$877$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$878$	0.831254	−	2.00682i	0.831254	−	2.00682i
$879$		0			0
$880$		0			0
$881$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$881$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$882$	−1.93415	−	1.93415i	−1.93415	−	1.93415i
$883$	0.312869			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$883$	0.312869			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$887$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$888$			0.227313i			0.227313i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−3.85890	−	1.59841i	−3.85890	−	1.59841i
$895$		0			0
$896$	−0.507083	+	0.210041i	−0.507083	+	0.210041i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.13649			−1.13649
$901$	−1.76007	+	0.896802i	−1.76007	+	0.896802i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$907$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$908$		0			0
$909$	−1.91031	+	1.91031i	−1.91031	+	1.91031i
$910$		0			0
$911$	−0.965451	−	0.399903i	−0.965451	−	0.399903i	−0.156434	−	0.987688i	$-0.550000\pi$
$911$	−0.965451	−	0.399903i	−0.965451	−	0.399903i	−0.809017	+	0.587785i	$0.800000\pi$
$912$		0			0
$913$		0			0
$914$		−	1.06740i		−	1.06740i
$915$		0			0
$916$		0			0
$917$	−0.796180			−0.796180
$918$	−4.61580	+	0.363271i	−4.61580	+	0.363271i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−0.692796	−	1.67256i	−0.692796	−	1.67256i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i
$926$		0			0
$927$	−3.74919	+	3.74919i	−3.74919	+	3.74919i
$928$		0			0
$929$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	−0.951057	−	0.309017i	$-0.900000\pi$
$929$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	0.891007	+	0.453990i	$0.150000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.652583	−	1.57547i	0.652583	−	1.57547i	−0.156434	−	0.987688i	$-0.550000\pi$
$941$	0.652583	−	1.57547i	0.652583	−	1.57547i	0.809017	−	0.587785i	$-0.200000\pi$
$942$	−0.677510	−	0.280634i	−0.677510	−	0.280634i
$943$		0			0
$944$	1.41421	−	1.41421i	1.41421	−	1.41421i
$945$		0			0
$946$		0			0
$947$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$948$		−	1.15822i		−	1.15822i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	−0.0266146	−	0.338170i	−0.0266146	−	0.338170i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	4.88569	+	4.88569i	4.88569	+	4.88569i
$955$		0			0
$956$			0.726543i			0.726543i
$957$		0			0
$958$	−0.210041	+	0.507083i	−0.210041	+	0.507083i
$959$		0			0
$960$		0			0
$961$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$962$		0			0
$963$		0			0
$964$	−0.0553750	+	0.0229371i	−0.0553750	+	0.0229371i
$965$		0			0
$966$		0			0
$967$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$967$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$968$	0.726543			0.726543
$969$		0			0
$970$		0			0
$971$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$	0.845800	+	2.04194i	0.845800	+	2.04194i
$973$		0			0
$974$	2.00682	−	0.831254i	2.00682	−	0.831254i
$975$		0			0
$976$	1.19336	+	0.494307i	1.19336	+	0.494307i
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$			0.367799i			0.367799i
$983$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$983$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.930903			−0.930903
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.965451	−	0.399903i	0.965451	−	0.399903i	0.156434	−	0.987688i	$-0.450000\pi$
$991$	0.965451	−	0.399903i	0.965451	−	0.399903i	0.809017	+	0.587785i	$0.200000\pi$
$992$		0			0
$993$	−2.16547	−	0.896969i	−2.16547	−	0.896969i
$994$	−0.590234	+	0.590234i	−0.590234	+	0.590234i
$995$		0			0
$996$	0.995048	+	0.412163i	0.995048	+	0.412163i
$997$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$997$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$998$		0			0
$999$			0.618034i			0.618034i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	799.1.h.b.610.2	yes	16
17.8	even	8	inner	799.1.h.b.93.2	✓	16
47.46	odd	2	CM	799.1.h.b.610.2	yes	16
799.93	odd	8	inner	799.1.h.b.93.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.1.h.b.93.2	✓	16	17.8	even	8	inner
799.1.h.b.93.2	✓	16	799.93	odd	8	inner
799.1.h.b.610.2	yes	16	1.1	even	1	trivial
799.1.h.b.610.2	yes	16	47.46	odd	2	CM

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:	\( N \)	\(=\)	\( 799 = 17 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	799.h (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.831254 - 0.831254i) q^{2} +(0.763007 + 1.84206i) q^{3} +0.381966i q^{4} +(0.896969 - 2.16547i) q^{6} +(0.431351 + 0.178671i) q^{7} +(-0.513743 + 0.513743i) q^{8} +(-2.10391 + 2.10391i) q^{9} +O(q^{10})\)	\(q+(-0.831254 - 0.831254i) q^{2} +(0.763007 + 1.84206i) q^{3} +0.381966i q^{4} +(0.896969 - 2.16547i) q^{6} +(0.431351 + 0.178671i) q^{7} +(-0.513743 + 0.513743i) q^{8} +(-2.10391 + 2.10391i) q^{9} +(-0.703605 + 0.291443i) q^{12} +(-0.210041 - 0.507083i) q^{14} +1.23607 q^{16} +(-0.309017 + 0.951057i) q^{17} +3.49777 q^{18} +0.930903i q^{21} +(-1.33834 - 0.554357i) q^{24} +(0.707107 - 0.707107i) q^{25} +(-3.63877 - 1.50723i) q^{27} +(-0.0682464 + 0.164761i) q^{28} +(-0.513743 - 0.513743i) q^{32} +(1.04744 - 0.533698i) q^{34} +(-0.803622 - 0.803622i) q^{36} +(-0.0600500 - 0.144974i) q^{37} +(0.773817 - 0.773817i) q^{42} +1.00000i q^{47} +(0.943129 + 2.27692i) q^{48} +(-0.552967 - 0.552967i) q^{49} -1.17557 q^{50} +(-1.98769 + 0.156434i) q^{51} +(1.39680 + 1.39680i) q^{53} +(1.77185 + 4.27763i) q^{54} +(-0.313395 + 0.129812i) q^{56} +(1.14412 - 1.14412i) q^{59} +(0.965451 + 0.399903i) q^{61} +(-1.28343 + 0.531615i) q^{63} -0.381966i q^{64} +(-0.363271 - 0.118034i) q^{68} +(-0.581990 - 1.40505i) q^{71} -2.16174i q^{72} +(-0.0705930 + 0.170427i) q^{74} +(1.84206 + 0.763007i) q^{75} +(-0.581990 + 1.40505i) q^{79} -4.87749i q^{81} +(-1.00000 - 1.00000i) q^{83} -0.355573 q^{84} -1.17557i q^{89} +(0.831254 - 0.831254i) q^{94} +(0.554357 - 1.33834i) q^{96} +(1.57547 - 0.652583i) q^{97} +0.919311i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^3 - 4 * q^9	\( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71} - 4 q^{79} - 16 q^{83} + 32 q^{84}+O(q^{100}) \) 16 * q - 4 * q^3 - 4 * q^9 - 16 * q^16 + 4 * q^17 - 20 * q^24 - 4 * q^27 - 4 * q^28 + 4 * q^36 - 20 * q^42 + 24 * q^48 + 4 * q^49 - 16 * q^51 + 4 * q^53 + 20 * q^54 + 20 * q^56 + 4 * q^61 - 4 * q^63 - 4 * q^71 - 4 * q^79 - 16 * q^83 + 32 * q^84

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