Properties

Label 1225.321
Modulus $1225$
Conductor $1225$
Order $70$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([42,5]))
 
pari: [g,chi] = znchar(Mod(321,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.bl

\(\chi_{1225}(6,\cdot)\) \(\chi_{1225}(41,\cdot)\) \(\chi_{1225}(111,\cdot)\) \(\chi_{1225}(181,\cdot)\) \(\chi_{1225}(216,\cdot)\) \(\chi_{1225}(286,\cdot)\) \(\chi_{1225}(321,\cdot)\) \(\chi_{1225}(356,\cdot)\) \(\chi_{1225}(461,\cdot)\) \(\chi_{1225}(496,\cdot)\) \(\chi_{1225}(531,\cdot)\) \(\chi_{1225}(566,\cdot)\) \(\chi_{1225}(671,\cdot)\) \(\chi_{1225}(706,\cdot)\) \(\chi_{1225}(741,\cdot)\) \(\chi_{1225}(811,\cdot)\) \(\chi_{1225}(846,\cdot)\) \(\chi_{1225}(916,\cdot)\) \(\chi_{1225}(986,\cdot)\) \(\chi_{1225}(1021,\cdot)\) \(\chi_{1225}(1056,\cdot)\) \(\chi_{1225}(1091,\cdot)\) \(\chi_{1225}(1161,\cdot)\) \(\chi_{1225}(1196,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((1177,101)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1225 }(321, a) \) \(-1\)\(1\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{19}{70}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{13}{70}\right)\)\(e\left(\frac{53}{70}\right)\)\(e\left(\frac{29}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1225 }(321,a) \;\) at \(\;a = \) e.g. 2