Properties

Label 4025.827
Modulus $4025$
Conductor $575$
Order $20$
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(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,0,10]))
 
pari: [g,chi] = znchar(Mod(827,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(575\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{575}(252,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.bj

\(\chi_{4025}(22,\cdot)\) \(\chi_{4025}(183,\cdot)\) \(\chi_{4025}(827,\cdot)\) \(\chi_{4025}(988,\cdot)\) \(\chi_{4025}(2437,\cdot)\) \(\chi_{4025}(2598,\cdot)\) \(\chi_{4025}(3242,\cdot)\) \(\chi_{4025}(3403,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.120567015877601807005703449249267578125.1

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(827, a) \) \(1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(827,a) \;\) at \(\;a = \) e.g. 2