Properties

Label 4025.1791
Modulus $4025$
Conductor $4025$
Order $110$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,55,25]))
 
pari: [g,chi] = znchar(Mod(1791,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(4025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.cv

\(\chi_{4025}(111,\cdot)\) \(\chi_{4025}(181,\cdot)\) \(\chi_{4025}(286,\cdot)\) \(\chi_{4025}(356,\cdot)\) \(\chi_{4025}(566,\cdot)\) \(\chi_{4025}(636,\cdot)\) \(\chi_{4025}(741,\cdot)\) \(\chi_{4025}(881,\cdot)\) \(\chi_{4025}(916,\cdot)\) \(\chi_{4025}(986,\cdot)\) \(\chi_{4025}(1056,\cdot)\) \(\chi_{4025}(1091,\cdot)\) \(\chi_{4025}(1161,\cdot)\) \(\chi_{4025}(1371,\cdot)\) \(\chi_{4025}(1441,\cdot)\) \(\chi_{4025}(1546,\cdot)\) \(\chi_{4025}(1581,\cdot)\) \(\chi_{4025}(1686,\cdot)\) \(\chi_{4025}(1721,\cdot)\) \(\chi_{4025}(1791,\cdot)\) \(\chi_{4025}(1861,\cdot)\) \(\chi_{4025}(1896,\cdot)\) \(\chi_{4025}(1966,\cdot)\) \(\chi_{4025}(2246,\cdot)\) \(\chi_{4025}(2386,\cdot)\) \(\chi_{4025}(2491,\cdot)\) \(\chi_{4025}(2596,\cdot)\) \(\chi_{4025}(2666,\cdot)\) \(\chi_{4025}(2771,\cdot)\) \(\chi_{4025}(2981,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((2577,1151,3501)\) → \((e\left(\frac{1}{5}\right),-1,e\left(\frac{5}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(1791, a) \) \(1\)\(1\)\(e\left(\frac{36}{55}\right)\)\(e\left(\frac{59}{110}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{21}{110}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{27}{110}\right)\)\(e\left(\frac{93}{110}\right)\)\(e\left(\frac{53}{110}\right)\)\(e\left(\frac{34}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(1791,a) \;\) at \(\;a = \) e.g. 2