Properties

Label 4025.488
Modulus $4025$
Conductor $4025$
Order $660$
Real no
Primitive yes
Minimal yes
Parity odd

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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([627,550,30]))
 
pari: [g,chi] = znchar(Mod(488,4025))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.dp

\(\chi_{4025}(17,\cdot)\) \(\chi_{4025}(33,\cdot)\) \(\chi_{4025}(38,\cdot)\) \(\chi_{4025}(103,\cdot)\) \(\chi_{4025}(122,\cdot)\) \(\chi_{4025}(152,\cdot)\) \(\chi_{4025}(178,\cdot)\) \(\chi_{4025}(222,\cdot)\) \(\chi_{4025}(227,\cdot)\) \(\chi_{4025}(283,\cdot)\) \(\chi_{4025}(297,\cdot)\) \(\chi_{4025}(313,\cdot)\) \(\chi_{4025}(327,\cdot)\) \(\chi_{4025}(362,\cdot)\) \(\chi_{4025}(383,\cdot)\) \(\chi_{4025}(388,\cdot)\) \(\chi_{4025}(402,\cdot)\) \(\chi_{4025}(458,\cdot)\) \(\chi_{4025}(467,\cdot)\) \(\chi_{4025}(488,\cdot)\) \(\chi_{4025}(502,\cdot)\) \(\chi_{4025}(523,\cdot)\) \(\chi_{4025}(563,\cdot)\) \(\chi_{4025}(572,\cdot)\) \(\chi_{4025}(612,\cdot)\) \(\chi_{4025}(628,\cdot)\) \(\chi_{4025}(642,\cdot)\) \(\chi_{4025}(663,\cdot)\) \(\chi_{4025}(677,\cdot)\) \(\chi_{4025}(733,\cdot)\) ...

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

\((2577,1151,3501)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{5}{6}\right),e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(488, a) \) \(-1\)\(1\)\(e\left(\frac{467}{660}\right)\)\(e\left(\frac{139}{660}\right)\)\(e\left(\frac{137}{330}\right)\)\(e\left(\frac{101}{110}\right)\)\(e\left(\frac{27}{220}\right)\)\(e\left(\frac{139}{330}\right)\)\(e\left(\frac{311}{330}\right)\)\(e\left(\frac{413}{660}\right)\)\(e\left(\frac{41}{220}\right)\)\(e\left(\frac{137}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(488,a) \;\) at \(\;a = \) e.g. 2