Properties

Label 4025.698
Modulus $4025$
Conductor $4025$
Order $660$
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(660))
 
M = H._module
 
chi = DirichletCharacter(H, M([363,550,180]))
 
pari: [g,chi] = znchar(Mod(698,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(4025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(660\)
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.do

\(\chi_{4025}(3,\cdot)\) \(\chi_{4025}(12,\cdot)\) \(\chi_{4025}(52,\cdot)\) \(\chi_{4025}(73,\cdot)\) \(\chi_{4025}(87,\cdot)\) \(\chi_{4025}(108,\cdot)\) \(\chi_{4025}(117,\cdot)\) \(\chi_{4025}(173,\cdot)\) \(\chi_{4025}(187,\cdot)\) \(\chi_{4025}(192,\cdot)\) \(\chi_{4025}(213,\cdot)\) \(\chi_{4025}(248,\cdot)\) \(\chi_{4025}(262,\cdot)\) \(\chi_{4025}(278,\cdot)\) \(\chi_{4025}(292,\cdot)\) \(\chi_{4025}(348,\cdot)\) \(\chi_{4025}(353,\cdot)\) \(\chi_{4025}(397,\cdot)\) \(\chi_{4025}(423,\cdot)\) \(\chi_{4025}(453,\cdot)\) \(\chi_{4025}(472,\cdot)\) \(\chi_{4025}(537,\cdot)\) \(\chi_{4025}(542,\cdot)\) \(\chi_{4025}(558,\cdot)\) \(\chi_{4025}(577,\cdot)\) \(\chi_{4025}(633,\cdot)\) \(\chi_{4025}(647,\cdot)\) \(\chi_{4025}(698,\cdot)\) \(\chi_{4025}(703,\cdot)\) \(\chi_{4025}(717,\cdot)\) ...

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: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

Values on generators

\((2577,1151,3501)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{5}{6}\right),e\left(\frac{3}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(698, a) \) \(1\)\(1\)\(e\left(\frac{503}{660}\right)\)\(e\left(\frac{31}{660}\right)\)\(e\left(\frac{173}{330}\right)\)\(e\left(\frac{89}{110}\right)\)\(e\left(\frac{63}{220}\right)\)\(e\left(\frac{31}{330}\right)\)\(e\left(\frac{97}{165}\right)\)\(e\left(\frac{377}{660}\right)\)\(e\left(\frac{169}{220}\right)\)\(e\left(\frac{8}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(698,a) \;\) at \(\;a = \) e.g. 2