Properties

Label 4025.208
Modulus $4025$
Conductor $175$
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(4025, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,50,0]))
 
pari: [g,chi] = znchar(Mod(208,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{175}(33,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.cj

\(\chi_{4025}(47,\cdot)\) \(\chi_{4025}(208,\cdot)\) \(\chi_{4025}(852,\cdot)\) \(\chi_{4025}(1013,\cdot)\) \(\chi_{4025}(1312,\cdot)\) \(\chi_{4025}(1473,\cdot)\) \(\chi_{4025}(2117,\cdot)\) \(\chi_{4025}(2278,\cdot)\) \(\chi_{4025}(2462,\cdot)\) \(\chi_{4025}(2623,\cdot)\) \(\chi_{4025}(2922,\cdot)\) \(\chi_{4025}(3083,\cdot)\) \(\chi_{4025}(3267,\cdot)\) \(\chi_{4025}(3428,\cdot)\) \(\chi_{4025}(3727,\cdot)\) \(\chi_{4025}(3888,\cdot)\)

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

\((2577,1151,3501)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{5}{6}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(208, a) \) \(1\)\(1\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{4}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(208,a) \;\) at \(\;a = \) e.g. 2