Properties

Label 4025.218
Modulus $4025$
Conductor $115$
Order $44$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,0,18]))
 
pari: [g,chi] = znchar(Mod(218,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(103,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.cc

\(\chi_{4025}(43,\cdot)\) \(\chi_{4025}(57,\cdot)\) \(\chi_{4025}(218,\cdot)\) \(\chi_{4025}(582,\cdot)\) \(\chi_{4025}(743,\cdot)\) \(\chi_{4025}(757,\cdot)\) \(\chi_{4025}(918,\cdot)\) \(\chi_{4025}(1282,\cdot)\) \(\chi_{4025}(1443,\cdot)\) \(\chi_{4025}(2682,\cdot)\) \(\chi_{4025}(2843,\cdot)\) \(\chi_{4025}(2857,\cdot)\) \(\chi_{4025}(3018,\cdot)\) \(\chi_{4025}(3032,\cdot)\) \(\chi_{4025}(3193,\cdot)\) \(\chi_{4025}(3207,\cdot)\) \(\chi_{4025}(3368,\cdot)\) \(\chi_{4025}(3557,\cdot)\) \(\chi_{4025}(3718,\cdot)\) \(\chi_{4025}(3907,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: \(\Q(\zeta_{115})^+\)

Values on generators

\((2577,1151,3501)\) → \((-i,1,e\left(\frac{9}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(218, a) \) \(1\)\(1\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{3}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(218,a) \;\) at \(\;a = \) e.g. 2