Properties

Label 2593.11
Modulus $2593$
Conductor $2593$
Order $2592$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2593, base_ring=CyclotomicField(2592))
 
M = H._module
 
chi = DirichletCharacter(H, M([533]))
 
pari: [g,chi] = znchar(Mod(11,2593))
 

Basic properties

Modulus: \(2593\)
Conductor: \(2593\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2592\)
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 2593.bd

\(\chi_{2593}(7,\cdot)\) \(\chi_{2593}(10,\cdot)\) \(\chi_{2593}(11,\cdot)\) \(\chi_{2593}(13,\cdot)\) \(\chi_{2593}(14,\cdot)\) \(\chi_{2593}(15,\cdot)\) \(\chi_{2593}(20,\cdot)\) \(\chi_{2593}(21,\cdot)\) \(\chi_{2593}(26,\cdot)\) \(\chi_{2593}(29,\cdot)\) \(\chi_{2593}(30,\cdot)\) \(\chi_{2593}(39,\cdot)\) \(\chi_{2593}(44,\cdot)\) \(\chi_{2593}(45,\cdot)\) \(\chi_{2593}(46,\cdot)\) \(\chi_{2593}(56,\cdot)\) \(\chi_{2593}(61,\cdot)\) \(\chi_{2593}(66,\cdot)\) \(\chi_{2593}(69,\cdot)\) \(\chi_{2593}(80,\cdot)\) \(\chi_{2593}(83,\cdot)\) \(\chi_{2593}(84,\cdot)\) \(\chi_{2593}(85,\cdot)\) \(\chi_{2593}(88,\cdot)\) \(\chi_{2593}(89,\cdot)\) \(\chi_{2593}(92,\cdot)\) \(\chi_{2593}(97,\cdot)\) \(\chi_{2593}(99,\cdot)\) \(\chi_{2593}(104,\cdot)\) \(\chi_{2593}(109,\cdot)\) ...

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: $\Q(\zeta_{2592})$
Fixed field: Number field defined by a degree 2592 polynomial (not computed)

Values on generators

\(7\) → \(e\left(\frac{533}{2592}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2593 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{46}{81}\right)\)\(e\left(\frac{593}{648}\right)\)\(e\left(\frac{11}{81}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{313}{648}\right)\)\(e\left(\frac{533}{2592}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{269}{324}\right)\)\(e\left(\frac{1229}{2592}\right)\)\(e\left(\frac{1561}{2592}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2593 }(11,a) \;\) at \(\;a = \) e.g. 2