Properties

Label 3150.2059
Modulus $3150$
Conductor $225$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,21,0]))
 
pari: [g,chi] = znchar(Mod(2059,3150))
 

Basic properties

Modulus: \(3150\)
Conductor: \(225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{225}(34,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3150.dq

\(\chi_{3150}(169,\cdot)\) \(\chi_{3150}(589,\cdot)\) \(\chi_{3150}(1219,\cdot)\) \(\chi_{3150}(1429,\cdot)\) \(\chi_{3150}(2059,\cdot)\) \(\chi_{3150}(2479,\cdot)\) \(\chi_{3150}(2689,\cdot)\) \(\chi_{3150}(3109,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.5399088047333990303844331037907977588474750518798828125.1

Values on generators

\((2801,127,451)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3150 }(2059, a) \) \(1\)\(1\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3150 }(2059,a) \;\) at \(\;a = \) e.g. 2