LMFDB - Embedded newform 2005.1.g.b.1202.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,1,Mod(1202,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 2]))

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

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

chi := DirichletCharacter("2005.1202");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2005.g (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:	$1.00062535033$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of 20.0.8181810539473601738391143798828125.1

Embedding invariants

Embedding label			1202.2
Root			$0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	2005.1202
Dual form			2005.1.g.b.1603.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.642040 - 0.642040i) q^{2} -0.175571i q^{4} +(0.309017 + 0.951057i) q^{5} +(0.221232 + 0.221232i) q^{7} +(-0.754763 + 0.754763i) q^{8} -1.00000i q^{9} +O(q^{10})$	\(q+(-0.642040 - 0.642040i) q^{2} -0.175571i q^{4} +(0.309017 + 0.951057i) q^{5} +(0.221232 + 0.221232i) q^{7} +(-0.754763 + 0.754763i) q^{8} -1.00000i q^{9} +(0.412215 - 0.809017i) q^{10} +1.17557 q^{11} -0.284079i q^{14} +0.793604 q^{16} +(-0.642040 + 0.642040i) q^{18} +(0.166977 - 0.0542543i) q^{20} +(-0.754763 - 0.754763i) q^{22} +(-0.809017 + 0.587785i) q^{25} +(0.0388418 - 0.0388418i) q^{28} +0.618034i q^{29} +(0.245237 + 0.245237i) q^{32} +(-0.142040 + 0.278768i) q^{35} -0.175571 q^{36} +(-0.951057 - 0.484587i) q^{40} +1.90211 q^{41} +(0.221232 - 0.221232i) q^{43} -0.206396i q^{44} +(0.951057 - 0.309017i) q^{45} +(0.642040 + 0.642040i) q^{47} -0.902113i q^{49} +(0.896802 + 0.142040i) q^{50} +(0.363271 + 1.11803i) q^{55} -0.333955 q^{56} +(0.396802 - 0.396802i) q^{58} +(0.221232 - 0.221232i) q^{63} -1.10851i q^{64} +(0.270175 - 0.0877853i) q^{70} +(0.754763 + 0.754763i) q^{72} +(0.642040 - 0.642040i) q^{73} +(0.260074 + 0.260074i) q^{77} +(0.245237 + 0.754763i) q^{80} -1.00000 q^{81} +(-1.22123 - 1.22123i) q^{82} +(1.00000 - 1.00000i) q^{83} -0.284079 q^{86} +(-0.887277 + 0.887277i) q^{88} +1.17557i q^{89} +(-0.809017 - 0.412215i) q^{90} -0.824429i q^{94} +(-0.579192 + 0.579192i) q^{98} -1.17557i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 8 * q - 2 * q^2 - 2 * q^5 + 2 * q^7	$ 8 q - 2 q^{2} - 2 q^{5} + 2 q^{7} + 8 q^{10} - 12 q^{16} - 2 q^{18} - 2 q^{25} - 12 q^{28} + 8 q^{32} + 2 q^{35} + 8 q^{36} + 2 q^{43} + 2 q^{47} - 2 q^{50} - 6 q^{58} + 2 q^{63} + 10 q^{70} + 2 q^{73} - 10 q^{77} + 8 q^{80} - 8 q^{81} - 10 q^{82} + 8 q^{83} + 4 q^{86} - 10 q^{88} - 2 q^{90} - 8 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 - 2 * q^5 + 2 * q^7 + 8 * q^10 - 12 * q^16 - 2 * q^18 - 2 * q^25 - 12 * q^28 + 8 * q^32 + 2 * q^35 + 8 * q^36 + 2 * q^43 + 2 * q^47 - 2 * q^50 - 6 * q^58 + 2 * q^63 + 10 * q^70 + 2 * q^73 - 10 * q^77 + 8 * q^80 - 8 * q^81 - 10 * q^82 + 8 * q^83 + 4 * q^86 - 10 * q^88 - 2 * q^90 - 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2005\mathbb{Z}\right)^\times$.

$n$	$402$	$1206$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$

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 $

$2$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$2$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		−	0.175571i		−	0.175571i
$5$	0.309017	+	0.951057i	0.309017	+	0.951057i
$5$	0.309017	+	0.951057i	0.309017	+	0.951057i
$6$		0			0
$7$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$7$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$8$	−0.754763	+	0.754763i	−0.754763	+	0.754763i
$9$		−	1.00000i		−	1.00000i
$10$	0.412215	−	0.809017i	0.412215	−	0.809017i
$11$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$11$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		−	0.284079i		−	0.284079i
$15$		0			0
$16$	0.793604			0.793604
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	−0.642040	+	0.642040i	−0.642040	+	0.642040i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	0.166977	−	0.0542543i	0.166977	−	0.0542543i
$21$		0			0
$22$	−0.754763	−	0.754763i	−0.754763	−	0.754763i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$26$		0			0
$27$		0			0
$28$	0.0388418	−	0.0388418i	0.0388418	−	0.0388418i
$29$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$29$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.245237	+	0.245237i	0.245237	+	0.245237i
$33$		0			0
$34$		0			0
$35$	−0.142040	+	0.278768i	−0.142040	+	0.278768i
$36$	−0.175571			−0.175571
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$	−0.951057	−	0.484587i	−0.951057	−	0.484587i
$41$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$41$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$	0.221232	−	0.221232i	0.221232	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$43$	0.221232	−	0.221232i	0.221232	−	0.221232i	0.809017	+	0.587785i	$0.200000\pi$
$44$		−	0.206396i		−	0.206396i
$45$	0.951057	−	0.309017i	0.951057	−	0.309017i
$46$		0			0
$47$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$47$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$48$		0			0
$49$		−	0.902113i		−	0.902113i
$50$	0.896802	+	0.142040i	0.896802	+	0.142040i
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$	0.363271	+	1.11803i	0.363271	+	1.11803i
$56$	−0.333955			−0.333955
$57$		0			0
$58$	0.396802	−	0.396802i	0.396802	−	0.396802i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	0.221232	−	0.221232i	0.221232	−	0.221232i
$64$		−	1.10851i		−	1.10851i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$	0.270175	−	0.0877853i	0.270175	−	0.0877853i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.754763	+	0.754763i	0.754763	+	0.754763i
$73$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$73$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.260074	+	0.260074i	0.260074	+	0.260074i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	0.245237	+	0.754763i	0.245237	+	0.754763i
$81$	−1.00000			−1.00000
$82$	−1.22123	−	1.22123i	−1.22123	−	1.22123i
$83$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$83$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$84$		0			0
$85$		0			0
$86$	−0.284079			−0.284079
$87$		0			0
$88$	−0.887277	+	0.887277i	−0.887277	+	0.887277i
$89$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$89$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$90$	−0.809017	−	0.412215i	−0.809017	−	0.412215i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		−	0.824429i		−	0.824429i
$95$		0			0
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	−0.579192	+	0.579192i	−0.579192	+	0.579192i
$99$		−	1.17557i		−	1.17557i
$100$	0.103198	+	0.142040i	0.103198	+	0.142040i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$103$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$109$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$110$	0.484587	−	0.951057i	0.484587	−	0.951057i
$111$		0			0
$112$	0.175571	+	0.175571i	0.175571	+	0.175571i
$113$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$113$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	0.587785	+	0.809017i	$0.300000\pi$
$114$		0			0
$115$		0			0
$116$	0.108509			0.108509
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.381966			0.381966
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$126$	−0.284079			−0.284079
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−0.466469	+	0.466469i	−0.466469	+	0.466469i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	0.0489435	+	0.0249380i	0.0489435	+	0.0249380i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	0.793604i		−	0.793604i
$145$	−0.587785	+	0.190983i	−0.587785	+	0.190983i
$146$	−0.824429			−0.824429
$147$		0			0
$148$		0			0
$149$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$149$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$150$		0			0
$151$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$151$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$152$		0			0
$153$		0			0
$154$		−	0.333955i		−	0.333955i
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$	−0.157452	+	0.309017i	−0.157452	+	0.309017i
$161$		0			0
$162$	0.642040	+	0.642040i	0.642040	+	0.642040i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		−	0.333955i		−	0.333955i
$165$		0			0
$166$	−1.28408			−1.28408
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	1.00000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$	−0.0388418	−	0.0388418i	−0.0388418	−	0.0388418i
$173$	0.221232	−	0.221232i	0.221232	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$173$	0.221232	−	0.221232i	0.221232	−	0.221232i	0.809017	+	0.587785i	$0.200000\pi$
$174$		0			0
$175$	−0.309017	−	0.0489435i	−0.309017	−	0.0489435i
$176$	0.932938			0.932938
$177$		0			0
$178$	0.754763	−	0.754763i	0.754763	−	0.754763i
$179$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$179$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$180$	−0.0542543	−	0.166977i	−0.0542543	−	0.166977i
$181$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$181$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.112723	−	0.112723i	0.112723	−	0.112723i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	−0.158384			−0.158384
$197$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$197$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$198$	−0.754763	+	0.754763i	−0.754763	+	0.754763i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	0.166977	−	1.05425i	0.166977	−	1.05425i
$201$		0			0
$202$		0			0
$203$	−0.136729	+	0.136729i	−0.136729	+	0.136729i
$204$		0			0
$205$	0.587785	+	1.80902i	0.587785	+	1.80902i
$206$	1.61803			1.61803
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.278768	+	0.142040i	0.278768	+	0.142040i
$216$		0			0
$217$		0			0
$218$	−1.03884	+	1.03884i	−1.03884	+	1.03884i
$219$		0			0
$220$	0.196294	−	0.0637797i	0.196294	−	0.0637797i
$221$		0			0
$222$		0			0
$223$	−1.39680	+	1.39680i	−1.39680	+	1.39680i	−0.587785	+	0.809017i	$0.700000\pi$
$223$	−1.39680	+	1.39680i	−1.39680	+	1.39680i	−0.809017	+	0.587785i	$0.800000\pi$
$224$			0.108509i			0.108509i
$225$	0.587785	+	0.809017i	0.587785	+	0.809017i
$226$	0.284079			0.284079
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$229$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$230$		0			0
$231$		0			0
$232$	−0.466469	−	0.466469i	−0.466469	−	0.466469i
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$	−0.412215	+	0.809017i	−0.412215	+	0.809017i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$239$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$240$		0			0
$241$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$241$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$242$	−0.245237	−	0.245237i	−0.245237	−	0.245237i
$243$		0			0
$244$		0			0
$245$	0.857960	−	0.278768i	0.857960	−	0.278768i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.142040	+	0.896802i	0.142040	+	0.896802i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−0.0388418	−	0.0388418i	−0.0388418	−	0.0388418i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.509525			−0.509525
$257$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$257$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.618034			0.618034
$262$		0			0
$263$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$263$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$		0			0
$280$	−0.103198	−	0.317610i	−0.103198	−	0.317610i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.420808	+	0.420808i	0.420808	+	0.420808i
$288$	0.245237	−	0.245237i	0.245237	−	0.245237i
$289$			1.00000i			1.00000i
$290$	0.500000	+	0.254763i	0.500000	+	0.254763i
$291$		0			0
$292$	−0.112723	−	0.112723i	−0.112723	−	0.112723i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−0.754763	+	0.754763i	−0.754763	+	0.754763i
$299$		0			0
$300$		0			0
$301$	0.0978870			0.0978870
$302$	1.22123	+	1.22123i	1.22123	+	1.22123i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$307$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$308$	0.0456612	−	0.0456612i	0.0456612	−	0.0456612i
$309$		0			0
$310$		0			0
$311$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$311$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$313$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	0.587785	+	0.809017i	$0.300000\pi$
$314$		0			0
$315$	0.278768	+	0.142040i	0.278768	+	0.142040i
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$			0.726543i			0.726543i
$320$	1.05425	−	0.342548i	1.05425	−	0.342548i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			0.175571i			0.175571i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−1.43564	+	1.43564i	−1.43564	+	1.43564i
$329$			0.284079i			0.284079i
$330$		0			0
$331$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$331$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$332$	−0.175571	−	0.175571i	−0.175571	−	0.175571i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$337$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$338$	−0.642040	+	0.642040i	−0.642040	+	0.642040i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.420808	−	0.420808i	0.420808	−	0.420808i
$344$			0.333955i			0.333955i
$345$		0			0
$346$	−0.284079			−0.284079
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	0.166977	+	0.229825i	0.166977	+	0.229825i
$351$		0			0
$352$	0.288294	+	0.288294i	0.288294	+	0.288294i
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$	0.206396			0.206396
$357$		0			0
$358$	0.396802	−	0.396802i	0.396802	−	0.396802i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−0.484587	+	0.951057i	−0.484587	+	0.951057i
$361$	−1.00000			−1.00000
$362$	−0.396802	−	0.396802i	−0.396802	−	0.396802i
$363$		0			0
$364$		0			0
$365$	0.809017	+	0.412215i	0.809017	+	0.412215i
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		−	1.90211i		−	1.90211i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$373$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$374$		0			0
$375$		0			0
$376$	−0.969175			−0.969175
$377$		0			0
$378$		0			0
$379$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$379$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$383$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$384$		0			0
$385$	−0.166977	+	0.327712i	−0.166977	+	0.327712i
$386$		0			0
$387$	−0.221232	−	0.221232i	−0.221232	−	0.221232i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	0.680881	+	0.680881i	0.680881	+	0.680881i
$393$		0			0
$394$		−	0.284079i		−	0.284079i
$395$		0			0
$396$	−0.206396			−0.206396
$397$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$397$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−0.642040	+	0.466469i	−0.642040	+	0.466469i
$401$	1.00000			1.00000
$401$	1.00000			1.00000
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$406$	0.175571			0.175571
$407$		0			0
$408$		0			0
$409$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$409$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$410$	0.784079	−	1.53884i	0.784079	−	1.53884i
$411$		0			0
$412$	0.221232	+	0.221232i	0.221232	+	0.221232i
$413$		0			0
$414$		0			0
$415$	1.26007	+	0.642040i	1.26007	+	0.642040i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$419$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	0.642040	−	0.642040i	0.642040	−	0.642040i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−0.0877853	−	0.270175i	−0.0877853	−	0.270175i
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.309017	−	0.951057i	$-0.400000\pi$
$433$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.951057	−	0.309017i	$-0.100000\pi$
$434$		0			0
$435$		0			0
$436$	−0.284079			−0.284079
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−1.11803	−	0.569667i	−1.11803	−	0.569667i
$441$	−0.902113			−0.902113
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$446$	1.79360			1.79360
$447$		0			0
$448$	0.245237	−	0.245237i	0.245237	−	0.245237i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	0.142040	−	0.896802i	0.142040	−	0.896802i
$451$	2.23607			2.23607
$452$	0.0388418	+	0.0388418i	0.0388418	+	0.0388418i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.809017	−	0.587785i	$-0.800000\pi$
$457$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.587785	−	0.809017i	$-0.700000\pi$
$458$	0.754763	−	0.754763i	0.754763	−	0.754763i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$			0.490475i			0.490475i
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$	0.784079	−	0.254763i	0.784079	−	0.254763i
$471$		0			0
$472$		0			0
$473$	0.260074	−	0.260074i	0.260074	−	0.260074i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.03884	−	1.03884i	1.03884	−	1.03884i
$479$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$479$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$480$		0			0
$481$		0			0
$482$	−0.396802	−	0.396802i	−0.396802	−	0.396802i
$483$		0			0
$484$		−	0.0670620i		−	0.0670620i
$485$		0			0
$486$		0			0
$487$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$487$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$	−0.729825	−	0.371864i	−0.729825	−	0.371864i
$491$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$491$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	1.11803	−	0.363271i	1.11803	−	0.363271i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	−0.103198	+	0.142040i	−0.103198	+	0.142040i
$501$		0			0
$502$		0			0
$503$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	−0.951057	−	0.309017i	$-0.900000\pi$
$503$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	0.309017	+	0.951057i	$0.400000\pi$
$504$			0.333955i			0.333955i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	0.284079			0.284079
$512$	0.793604	+	0.793604i	0.793604	+	0.793604i
$513$		0			0
$514$			1.28408i			1.28408i
$515$	−1.58779	−	0.809017i	−1.58779	−	0.809017i
$516$		0			0
$517$	0.754763	+	0.754763i	0.754763	+	0.754763i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−0.396802	−	0.396802i	−0.396802	−	0.396802i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$	1.61803			1.61803
$527$		0			0
$528$		0			0
$529$		−	1.00000i		−	1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1.06050i		−	1.06050i
$540$		0			0
$541$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$541$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.53884	−	0.500000i	1.53884	−	0.500000i
$546$		0			0
$547$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.951057	+	0.309017i	$0.100000\pi$
$547$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$548$		0			0
$549$		0			0
$550$	1.05425	+	0.166977i	1.05425	+	0.166977i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$557$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$558$		0			0
$559$		0			0
$560$	−0.112723	+	0.221232i	−0.112723	+	0.221232i
$561$		0			0
$562$		0			0
$563$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$563$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$564$		0			0
$565$	−0.278768	−	0.142040i	−0.278768	−	0.142040i
$566$		0			0
$567$	−0.221232	−	0.221232i	−0.221232	−	0.221232i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		−	0.540350i		−	0.540350i
$575$		0			0
$576$	−1.10851			−1.10851
$577$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$577$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$578$	0.642040	−	0.642040i	0.642040	−	0.642040i
$579$		0			0
$580$	0.0335310	+	0.103198i	0.0335310	+	0.103198i
$581$	0.442463			0.442463
$582$		0			0
$583$		0			0
$584$			0.969175i			0.969175i
$585$		0			0
$586$		0			0
$587$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$587$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$	−0.206396			−0.206396
$597$		0			0
$598$		0			0
$599$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$599$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$600$		0			0
$601$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$601$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$602$	−0.0628473	−	0.0628473i	−0.0628473	−	0.0628473i
$603$		0			0
$604$			0.333955i			0.333955i
$605$	0.118034	+	0.363271i	0.118034	+	0.363271i
$606$		0			0
$607$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.809017	−	0.587785i	$-0.800000\pi$
$607$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.587785	−	0.809017i	$-0.700000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$			1.61803i			1.61803i
$615$		0			0
$616$	−0.392588			−0.392588
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$619$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$620$		0			0
$621$		0			0
$622$	0.396802	+	0.396802i	0.396802	+	0.396802i
$623$	−0.260074	+	0.260074i	−0.260074	+	0.260074i
$624$		0			0
$625$	0.309017	−	0.951057i	0.309017	−	0.951057i
$626$	0.284079			0.284079
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−0.0877853	−	0.270175i	−0.0877853	−	0.270175i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.466469	−	0.466469i	0.466469	−	0.466469i
$639$		0			0
$640$	−0.587785	−	0.299492i	−0.587785	−	0.299492i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$643$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	0.754763	−	0.754763i	0.754763	−	0.754763i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$653$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$654$		0			0
$655$		0			0
$656$	1.50953			1.50953
$657$	−0.642040	−	0.642040i	−0.642040	−	0.642040i
$658$	0.182390	−	0.182390i	0.182390	−	0.182390i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	1.22123	+	1.22123i	1.22123	+	1.22123i
$663$		0			0
$664$			1.50953i			1.50953i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$			1.28408i			1.28408i
$675$		0			0
$676$	−0.175571			−0.175571
$677$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$677$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$	−0.540350			−0.540350
$687$		0			0
$688$	0.175571	−	0.175571i	0.175571	−	0.175571i
$689$		0			0
$690$		0			0
$691$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$691$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$692$	−0.0388418	−	0.0388418i	−0.0388418	−	0.0388418i
$693$	0.260074	−	0.260074i	0.260074	−	0.260074i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.00859303	+	0.0542543i	−0.00859303	+	0.0542543i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		−	1.30313i		−	1.30313i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$709$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$710$		0			0
$711$		0			0
$712$	−0.887277	−	0.887277i	−0.887277	−	0.887277i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.108509			0.108509
$717$		0			0
$718$		0			0
$719$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$719$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$720$	0.754763	−	0.245237i	0.754763	−	0.245237i
$721$	−0.557537			−0.557537
$722$	0.642040	+	0.642040i	0.642040	+	0.642040i
$723$		0			0
$724$		−	0.108509i		−	0.108509i
$725$	−0.363271	−	0.500000i	−0.363271	−	0.500000i
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$	−0.254763	−	0.784079i	−0.254763	−	0.784079i
$731$		0			0
$732$		0			0
$733$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$733$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−1.22123	+	1.22123i	−1.22123	+	1.22123i
$739$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$739$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	1.11803	−	0.363271i	1.11803	−	0.363271i
$746$	−0.824429			−0.824429
$747$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$751$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$752$	0.509525	+	0.509525i	0.509525	+	0.509525i
$753$		0			0
$754$		0			0
$755$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$756$		0			0
$757$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.951057	+	0.309017i	$0.100000\pi$
$757$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$758$	1.28408	−	1.28408i	1.28408	−	1.28408i
$759$		0			0
$760$		0			0
$761$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$761$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$762$		0			0
$763$	0.357960	−	0.357960i	0.357960	−	0.357960i
$764$		0			0
$765$		0			0
$766$	1.28408			1.28408
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$	0.317610	−	0.103198i	0.317610	−	0.103198i
$771$		0			0
$772$		0			0
$773$	0.221232	−	0.221232i	0.221232	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$773$	0.221232	−	0.221232i	0.221232	−	0.221232i	0.809017	+	0.587785i	$0.200000\pi$
$774$			0.284079i			0.284079i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	0.715921i		−	0.715921i
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	0.0388418	−	0.0388418i	0.0388418	−	0.0388418i
$789$		0			0
$790$		0			0
$791$	−0.0978870			−0.0978870
$792$	0.887277	+	0.887277i	0.887277	+	0.887277i
$793$		0			0
$794$			1.28408i			1.28408i
$795$		0			0
$796$		0			0
$797$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$797$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−0.342548	−	0.0542543i	−0.342548	−	0.0542543i
$801$	1.17557			1.17557
$802$	−0.642040	−	0.642040i	−0.642040	−	0.642040i
$803$	0.754763	−	0.754763i	0.754763	−	0.754763i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−0.412215	+	0.809017i	−0.412215	+	0.809017i
$811$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$811$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$812$	0.0240055	+	0.0240055i	0.0240055	+	0.0240055i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.03884	−	1.03884i	1.03884	−	1.03884i
$819$		0			0
$820$	0.317610	−	0.103198i	0.317610	−	0.103198i
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		−	1.90211i		−	1.90211i
$825$		0			0
$826$		0			0
$827$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$827$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$	−0.396802	−	1.22123i	−0.396802	−	1.22123i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	1.03884	−	1.03884i	1.03884	−	1.03884i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	0.618034			0.618034
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.951057	−	0.309017i	0.951057	−	0.309017i
$846$	−0.824429			−0.824429
$847$	0.0845030	+	0.0845030i	0.0845030	+	0.0845030i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.309017	−	0.951057i	$-0.400000\pi$
$853$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.951057	−	0.309017i	$-0.100000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$857$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$858$		0			0
$859$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$859$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$860$	0.0249380	−	0.0489435i	0.0249380	−	0.0489435i
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$	0.278768	+	0.142040i	0.278768	+	0.142040i
$866$	−1.61803			−1.61803
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	1.22123	+	1.22123i	1.22123	+	1.22123i
$873$		0			0
$874$		0			0
$875$	−0.0489435	−	0.309017i	−0.0489435	−	0.309017i
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$		0			0
$880$	0.288294	+	0.887277i	0.288294	+	0.887277i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.579192	+	0.579192i	0.579192	+	0.579192i
$883$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$883$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$	0.951057	+	0.484587i	0.951057	+	0.484587i
$891$	−1.17557			−1.17557
$892$	0.245237	+	0.245237i	0.245237	+	0.245237i
$893$		0			0
$894$		0			0
$895$	−0.587785	+	0.190983i	−0.587785	+	0.190983i
$896$	−0.206396			−0.206396
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.142040	−	0.103198i	0.142040	−	0.103198i
$901$		0			0
$902$	−1.43564	−	1.43564i	−1.43564	−	1.43564i
$903$		0			0
$904$		−	0.333955i		−	0.333955i
$905$	0.190983	+	0.587785i	0.190983	+	0.587785i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$911$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$912$		0			0
$913$	1.17557	−	1.17557i	1.17557	−	1.17557i
$914$			1.79360i			1.79360i
$915$		0			0
$916$	0.206396			0.206396
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.26007	+	1.26007i	1.26007	+	1.26007i
$928$	−0.151565	+	0.151565i	−0.151565	+	0.151565i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		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
$940$	0.142040	+	0.0723728i	0.142040	+	0.0723728i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−0.333955			−0.333955
$947$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$947$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$953$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$954$		0			0
$955$		0			0
$956$	0.284079			0.284079
$957$		0			0
$958$	−0.754763	+	0.754763i	−0.754763	+	0.754763i
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$		−	0.108509i		−	0.108509i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−0.288294	+	0.288294i	−0.288294	+	0.288294i
$969$		0			0
$970$		0			0
$971$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$971$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$972$		0			0
$973$		0			0
$974$		−	1.28408i		−	1.28408i
$975$		0			0
$976$		0			0
$977$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$977$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$978$		0			0
$979$			1.38197i			1.38197i
$980$	−0.0489435	−	0.150633i	−0.0489435	−	0.150633i
$981$	−1.61803			−1.61803
$982$	1.03884	+	1.03884i	1.03884	+	1.03884i
$983$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$983$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	0.587785	+	0.809017i	$0.300000\pi$
$984$		0			0
$985$	−0.142040	+	0.278768i	−0.142040	+	0.278768i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.951057	−	0.484587i	−0.951057	−	0.484587i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$997$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.1.g.b.1202.2	✓	8
5.3	odd	4	inner	2005.1.g.b.1603.2	yes	8
401.400	even	2	RM	2005.1.g.b.1202.2	✓	8
2005.1603	odd	4	inner	2005.1.g.b.1603.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.1.g.b.1202.2	✓	8	1.1	even	1	trivial
2005.1.g.b.1202.2	✓	8	401.400	even	2	RM
2005.1.g.b.1603.2	yes	8	5.3	odd	4	inner
2005.1.g.b.1603.2	yes	8	2005.1603	odd	4	inner

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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2005.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.642040 - 0.642040i) q^{2} -0.175571i q^{4} +(0.309017 + 0.951057i) q^{5} +(0.221232 + 0.221232i) q^{7} +(-0.754763 + 0.754763i) q^{8} -1.00000i q^{9} +O(q^{10})\)	\(q+(-0.642040 - 0.642040i) q^{2} -0.175571i q^{4} +(0.309017 + 0.951057i) q^{5} +(0.221232 + 0.221232i) q^{7} +(-0.754763 + 0.754763i) q^{8} -1.00000i q^{9} +(0.412215 - 0.809017i) q^{10} +1.17557 q^{11} -0.284079i q^{14} +0.793604 q^{16} +(-0.642040 + 0.642040i) q^{18} +(0.166977 - 0.0542543i) q^{20} +(-0.754763 - 0.754763i) q^{22} +(-0.809017 + 0.587785i) q^{25} +(0.0388418 - 0.0388418i) q^{28} +0.618034i q^{29} +(0.245237 + 0.245237i) q^{32} +(-0.142040 + 0.278768i) q^{35} -0.175571 q^{36} +(-0.951057 - 0.484587i) q^{40} +1.90211 q^{41} +(0.221232 - 0.221232i) q^{43} -0.206396i q^{44} +(0.951057 - 0.309017i) q^{45} +(0.642040 + 0.642040i) q^{47} -0.902113i q^{49} +(0.896802 + 0.142040i) q^{50} +(0.363271 + 1.11803i) q^{55} -0.333955 q^{56} +(0.396802 - 0.396802i) q^{58} +(0.221232 - 0.221232i) q^{63} -1.10851i q^{64} +(0.270175 - 0.0877853i) q^{70} +(0.754763 + 0.754763i) q^{72} +(0.642040 - 0.642040i) q^{73} +(0.260074 + 0.260074i) q^{77} +(0.245237 + 0.754763i) q^{80} -1.00000 q^{81} +(-1.22123 - 1.22123i) q^{82} +(1.00000 - 1.00000i) q^{83} -0.284079 q^{86} +(-0.887277 + 0.887277i) q^{88} +1.17557i q^{89} +(-0.809017 - 0.412215i) q^{90} -0.824429i q^{94} +(-0.579192 + 0.579192i) q^{98} -1.17557i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 8 * q - 2 * q^2 - 2 * q^5 + 2 * q^7	\( 8 q - 2 q^{2} - 2 q^{5} + 2 q^{7} + 8 q^{10} - 12 q^{16} - 2 q^{18} - 2 q^{25} - 12 q^{28} + 8 q^{32} + 2 q^{35} + 8 q^{36} + 2 q^{43} + 2 q^{47} - 2 q^{50} - 6 q^{58} + 2 q^{63} + 10 q^{70} + 2 q^{73} - 10 q^{77} + 8 q^{80} - 8 q^{81} - 10 q^{82} + 8 q^{83} + 4 q^{86} - 10 q^{88} - 2 q^{90} - 8 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 - 2 * q^5 + 2 * q^7 + 8 * q^10 - 12 * q^16 - 2 * q^18 - 2 * q^25 - 12 * q^28 + 8 * q^32 + 2 * q^35 + 8 * q^36 + 2 * q^43 + 2 * q^47 - 2 * q^50 - 6 * q^58 + 2 * q^63 + 10 * q^70 + 2 * q^73 - 10 * q^77 + 8 * q^80 - 8 * q^81 - 10 * q^82 + 8 * q^83 + 4 * q^86 - 10 * q^88 - 2 * q^90 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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