Properties

Label 7581.107
Modulus $7581$
Conductor $7581$
Order $114$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7581, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([57,38,89]))
 
pari: [g,chi] = znchar(Mod(107,7581))
 

Basic properties

Modulus: \(7581\)
Conductor: \(7581\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
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 7581.cz

\(\chi_{7581}(107,\cdot)\) \(\chi_{7581}(179,\cdot)\) \(\chi_{7581}(506,\cdot)\) \(\chi_{7581}(578,\cdot)\) \(\chi_{7581}(905,\cdot)\) \(\chi_{7581}(977,\cdot)\) \(\chi_{7581}(1304,\cdot)\) \(\chi_{7581}(1703,\cdot)\) \(\chi_{7581}(1775,\cdot)\) \(\chi_{7581}(2102,\cdot)\) \(\chi_{7581}(2174,\cdot)\) \(\chi_{7581}(2501,\cdot)\) \(\chi_{7581}(2573,\cdot)\) \(\chi_{7581}(2900,\cdot)\) \(\chi_{7581}(2972,\cdot)\) \(\chi_{7581}(3299,\cdot)\) \(\chi_{7581}(3371,\cdot)\) \(\chi_{7581}(3698,\cdot)\) \(\chi_{7581}(3770,\cdot)\) \(\chi_{7581}(4097,\cdot)\) \(\chi_{7581}(4169,\cdot)\) \(\chi_{7581}(4496,\cdot)\) \(\chi_{7581}(4568,\cdot)\) \(\chi_{7581}(4895,\cdot)\) \(\chi_{7581}(4967,\cdot)\) \(\chi_{7581}(5294,\cdot)\) \(\chi_{7581}(5366,\cdot)\) \(\chi_{7581}(5693,\cdot)\) \(\chi_{7581}(5765,\cdot)\) \(\chi_{7581}(6092,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 114 polynomial (not computed)

Values on generators

\((2528,6499,1807)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{89}{114}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(20\)
\( \chi_{ 7581 }(107, a) \) \(1\)\(1\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{17}{19}\right)\)\(e\left(\frac{11}{38}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{9}{38}\right)\)\(e\left(\frac{53}{114}\right)\)\(e\left(\frac{97}{114}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{31}{114}\right)\)\(e\left(\frac{7}{38}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7581 }(107,a) \;\) at \(\;a = \) e.g. 2