Properties

Label 1225.501
Modulus 12251225
Conductor 4949
Order 2121
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,40]))
 
pari: [g,chi] = znchar(Mod(501,1225))
 

Basic properties

Modulus: 12251225
Conductor: 4949
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 2121
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from χ49(11,)\chi_{49}(11,\cdot)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.x

χ1225(51,)\chi_{1225}(51,\cdot) χ1225(151,)\chi_{1225}(151,\cdot) χ1225(326,)\chi_{1225}(326,\cdot) χ1225(401,)\chi_{1225}(401,\cdot) χ1225(501,)\chi_{1225}(501,\cdot) χ1225(576,)\chi_{1225}(576,\cdot) χ1225(676,)\chi_{1225}(676,\cdot) χ1225(751,)\chi_{1225}(751,\cdot) χ1225(926,)\chi_{1225}(926,\cdot) χ1225(1026,)\chi_{1225}(1026,\cdot) χ1225(1101,)\chi_{1225}(1101,\cdot) χ1225(1201,)\chi_{1225}(1201,\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(ζ21)\Q(\zeta_{21})
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

(1177,101)(1177,101)(1,e(2021))(1,e\left(\frac{20}{21}\right))

First values

aa 1-1112233446688991111121213131616
χ1225(501,a) \chi_{ 1225 }(501, a) 1111e(1621)e\left(\frac{16}{21}\right)e(2021)e\left(\frac{20}{21}\right)e(1121)e\left(\frac{11}{21}\right)e(57)e\left(\frac{5}{7}\right)e(27)e\left(\frac{2}{7}\right)e(1921)e\left(\frac{19}{21}\right)e(221)e\left(\frac{2}{21}\right)e(1021)e\left(\frac{10}{21}\right)e(37)e\left(\frac{3}{7}\right)e(121)e\left(\frac{1}{21}\right)
sage: chi.jacobi_sum(n)
 
χ1225(501,a)   \chi_{ 1225 }(501,a) \; at   a=\;a = e.g. 2