Properties

Label 3150.113
Modulus $3150$
Conductor $225$
Order $60$
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(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,57,0]))
 
pari: [g,chi] = znchar(Mod(113,3150))
 

Basic properties

Modulus: \(3150\)
Conductor: \(225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{225}(113,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3150.eh

\(\chi_{3150}(113,\cdot)\) \(\chi_{3150}(533,\cdot)\) \(\chi_{3150}(617,\cdot)\) \(\chi_{3150}(1037,\cdot)\) \(\chi_{3150}(1163,\cdot)\) \(\chi_{3150}(1247,\cdot)\) \(\chi_{3150}(1373,\cdot)\) \(\chi_{3150}(1667,\cdot)\) \(\chi_{3150}(1877,\cdot)\) \(\chi_{3150}(2003,\cdot)\) \(\chi_{3150}(2297,\cdot)\) \(\chi_{3150}(2423,\cdot)\) \(\chi_{3150}(2633,\cdot)\) \(\chi_{3150}(2927,\cdot)\) \(\chi_{3150}(3053,\cdot)\) \(\chi_{3150}(3137,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((2801,127,451)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{19}{20}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3150 }(113, a) \) \(1\)\(1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3150 }(113,a) \;\) at \(\;a = \) e.g. 2