Properties

Label 1205.194
Modulus $1205$
Conductor $1205$
Order $40$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1205, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,39]))
 
pari: [g,chi] = znchar(Mod(194,1205))
 

Basic properties

Modulus: \(1205\)
Conductor: \(1205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
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 1205.bw

\(\chi_{1205}(79,\cdot)\) \(\chi_{1205}(194,\cdot)\) \(\chi_{1205}(214,\cdot)\) \(\chi_{1205}(289,\cdot)\) \(\chi_{1205}(434,\cdot)\) \(\chi_{1205}(509,\cdot)\) \(\chi_{1205}(529,\cdot)\) \(\chi_{1205}(644,\cdot)\) \(\chi_{1205}(764,\cdot)\) \(\chi_{1205}(784,\cdot)\) \(\chi_{1205}(839,\cdot)\) \(\chi_{1205}(959,\cdot)\) \(\chi_{1205}(969,\cdot)\) \(\chi_{1205}(1089,\cdot)\) \(\chi_{1205}(1144,\cdot)\) \(\chi_{1205}(1164,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((242,971)\) → \((-1,e\left(\frac{39}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1205 }(194, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{19}{20}\right)\)\(-1\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{19}{40}\right)\)\(i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{13}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1205 }(194,a) \;\) at \(\;a = \) e.g. 2