Properties

Label 2675.814
Modulus $2675$
Conductor $2675$
Order $530$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2675, base_ring=CyclotomicField(530))
 
M = H._module
 
chi = DirichletCharacter(H, M([159,305]))
 
pari: [g,chi] = znchar(Mod(814,2675))
 

Basic properties

Modulus: \(2675\)
Conductor: \(2675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(530\)
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 2675.v

\(\chi_{2675}(54,\cdot)\) \(\chi_{2675}(59,\cdot)\) \(\chi_{2675}(84,\cdot)\) \(\chi_{2675}(94,\cdot)\) \(\chi_{2675}(104,\cdot)\) \(\chi_{2675}(109,\cdot)\) \(\chi_{2675}(114,\cdot)\) \(\chi_{2675}(129,\cdot)\) \(\chi_{2675}(139,\cdot)\) \(\chi_{2675}(179,\cdot)\) \(\chi_{2675}(184,\cdot)\) \(\chi_{2675}(189,\cdot)\) \(\chi_{2675}(204,\cdot)\) \(\chi_{2675}(219,\cdot)\) \(\chi_{2675}(229,\cdot)\) \(\chi_{2675}(234,\cdot)\) \(\chi_{2675}(259,\cdot)\) \(\chi_{2675}(264,\cdot)\) \(\chi_{2675}(269,\cdot)\) \(\chi_{2675}(279,\cdot)\) \(\chi_{2675}(284,\cdot)\) \(\chi_{2675}(294,\cdot)\) \(\chi_{2675}(309,\cdot)\) \(\chi_{2675}(329,\cdot)\) \(\chi_{2675}(339,\cdot)\) \(\chi_{2675}(359,\cdot)\) \(\chi_{2675}(364,\cdot)\) \(\chi_{2675}(379,\cdot)\) \(\chi_{2675}(384,\cdot)\) \(\chi_{2675}(389,\cdot)\) ...

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: $\Q(\zeta_{265})$
Fixed field: Number field defined by a degree 530 polynomial (not computed)

Values on generators

\((1927,751)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{61}{106}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 2675 }(814, a) \) \(-1\)\(1\)\(e\left(\frac{232}{265}\right)\)\(e\left(\frac{203}{530}\right)\)\(e\left(\frac{199}{265}\right)\)\(e\left(\frac{137}{530}\right)\)\(e\left(\frac{13}{53}\right)\)\(e\left(\frac{166}{265}\right)\)\(e\left(\frac{203}{265}\right)\)\(e\left(\frac{122}{265}\right)\)\(e\left(\frac{71}{530}\right)\)\(e\left(\frac{401}{530}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2675 }(814,a) \;\) at \(\;a = \) e.g. 2