Properties

Label 2624.115
Modulus $2624$
Conductor $2624$
Order $80$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2624, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,75,36]))
 
pari: [g,chi] = znchar(Mod(115,2624))
 

Basic properties

Modulus: \(2624\)
Conductor: \(2624\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
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 2624.en

\(\chi_{2624}(115,\cdot)\) \(\chi_{2624}(203,\cdot)\) \(\chi_{2624}(251,\cdot)\) \(\chi_{2624}(267,\cdot)\) \(\chi_{2624}(307,\cdot)\) \(\chi_{2624}(323,\cdot)\) \(\chi_{2624}(371,\cdot)\) \(\chi_{2624}(459,\cdot)\) \(\chi_{2624}(771,\cdot)\) \(\chi_{2624}(859,\cdot)\) \(\chi_{2624}(907,\cdot)\) \(\chi_{2624}(923,\cdot)\) \(\chi_{2624}(963,\cdot)\) \(\chi_{2624}(979,\cdot)\) \(\chi_{2624}(1027,\cdot)\) \(\chi_{2624}(1115,\cdot)\) \(\chi_{2624}(1427,\cdot)\) \(\chi_{2624}(1515,\cdot)\) \(\chi_{2624}(1563,\cdot)\) \(\chi_{2624}(1579,\cdot)\) \(\chi_{2624}(1619,\cdot)\) \(\chi_{2624}(1635,\cdot)\) \(\chi_{2624}(1683,\cdot)\) \(\chi_{2624}(1771,\cdot)\) \(\chi_{2624}(2083,\cdot)\) \(\chi_{2624}(2171,\cdot)\) \(\chi_{2624}(2219,\cdot)\) \(\chi_{2624}(2235,\cdot)\) \(\chi_{2624}(2275,\cdot)\) \(\chi_{2624}(2291,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((575,1477,129)\) → \((-1,e\left(\frac{15}{16}\right),e\left(\frac{9}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2624 }(115, a) \) \(-1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{67}{80}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{43}{80}\right)\)\(e\left(\frac{1}{80}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{80}\right)\)\(e\left(\frac{39}{80}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2624 }(115,a) \;\) at \(\;a = \) e.g. 2