Properties

Label 2001.80
Modulus 20012001
Conductor 20012001
Order 154154
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,63,143]))
 
pari: [g,chi] = znchar(Mod(80,2001))
 

Basic properties

Modulus: 20012001
Conductor: 20012001
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 154154
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2001.br

χ2001(5,)\chi_{2001}(5,\cdot) χ2001(38,)\chi_{2001}(38,\cdot) χ2001(80,)\chi_{2001}(80,\cdot) χ2001(122,)\chi_{2001}(122,\cdot) χ2001(125,)\chi_{2001}(125,\cdot) χ2001(149,)\chi_{2001}(149,\cdot) χ2001(158,)\chi_{2001}(158,\cdot) χ2001(212,)\chi_{2001}(212,\cdot) χ2001(245,)\chi_{2001}(245,\cdot) χ2001(296,)\chi_{2001}(296,\cdot) χ2001(332,)\chi_{2001}(332,\cdot) χ2001(341,)\chi_{2001}(341,\cdot) χ2001(383,)\chi_{2001}(383,\cdot) χ2001(410,)\chi_{2001}(410,\cdot) χ2001(419,)\chi_{2001}(419,\cdot) χ2001(428,)\chi_{2001}(428,\cdot) χ2001(470,)\chi_{2001}(470,\cdot) χ2001(497,)\chi_{2001}(497,\cdot) χ2001(527,)\chi_{2001}(527,\cdot) χ2001(557,)\chi_{2001}(557,\cdot) χ2001(701,)\chi_{2001}(701,\cdot) χ2001(734,)\chi_{2001}(734,\cdot) χ2001(776,)\chi_{2001}(776,\cdot) χ2001(845,)\chi_{2001}(845,\cdot) χ2001(908,)\chi_{2001}(908,\cdot) χ2001(941,)\chi_{2001}(941,\cdot) χ2001(950,)\chi_{2001}(950,\cdot) χ2001(962,)\chi_{2001}(962,\cdot) χ2001(1019,)\chi_{2001}(1019,\cdot) χ2001(1049,)\chi_{2001}(1049,\cdot) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: Q(ζ77)\Q(\zeta_{77})
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

(668,1132,553)(668,1132,553)(1,e(922),e(1314))(-1,e\left(\frac{9}{22}\right),e\left(\frac{13}{14}\right))

First values

aa 1-111224455778810101111131314141616
χ2001(80,a) \chi_{ 2001 }(80, a) 1111e(1977)e\left(\frac{19}{77}\right)e(3877)e\left(\frac{38}{77}\right)e(2677)e\left(\frac{26}{77}\right)e(141154)e\left(\frac{141}{154}\right)e(5777)e\left(\frac{57}{77}\right)e(4577)e\left(\frac{45}{77}\right)e(61154)e\left(\frac{61}{154}\right)e(3477)e\left(\frac{34}{77}\right)e(25154)e\left(\frac{25}{154}\right)e(7677)e\left(\frac{76}{77}\right)
sage: chi.jacobi_sum(n)
 
χ2001(80,a)   \chi_{ 2001 }(80,a) \; at   a=\;a = e.g. 2