Properties

Label 1225.1119
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([63,25]))
 
pari: [g,chi] = znchar(Mod(1119,1225))
 

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.bk

\(\chi_{1225}(34,\cdot)\) \(\chi_{1225}(69,\cdot)\) \(\chi_{1225}(104,\cdot)\) \(\chi_{1225}(139,\cdot)\) \(\chi_{1225}(209,\cdot)\) \(\chi_{1225}(279,\cdot)\) \(\chi_{1225}(314,\cdot)\) \(\chi_{1225}(384,\cdot)\) \(\chi_{1225}(419,\cdot)\) \(\chi_{1225}(454,\cdot)\) \(\chi_{1225}(559,\cdot)\) \(\chi_{1225}(594,\cdot)\) \(\chi_{1225}(629,\cdot)\) \(\chi_{1225}(664,\cdot)\) \(\chi_{1225}(769,\cdot)\) \(\chi_{1225}(804,\cdot)\) \(\chi_{1225}(839,\cdot)\) \(\chi_{1225}(909,\cdot)\) \(\chi_{1225}(944,\cdot)\) \(\chi_{1225}(1014,\cdot)\) \(\chi_{1225}(1084,\cdot)\) \(\chi_{1225}(1119,\cdot)\) \(\chi_{1225}(1154,\cdot)\) \(\chi_{1225}(1189,\cdot)\)

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

\((1177,101)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1225 }(1119, a) \) \(-1\)\(1\)\(e\left(\frac{13}{70}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{59}{70}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{26}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1225 }(1119,a) \;\) at \(\;a = \) e.g. 2