Properties

Label 5929.78
Modulus $5929$
Conductor $5929$
Order $77$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5929, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([66,112]))
 
pari: [g,chi] = znchar(Mod(78,5929))
 

Basic properties

Modulus: \(5929\)
Conductor: \(5929\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(77\)
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 5929.bl

\(\chi_{5929}(78,\cdot)\) \(\chi_{5929}(155,\cdot)\) \(\chi_{5929}(232,\cdot)\) \(\chi_{5929}(309,\cdot)\) \(\chi_{5929}(386,\cdot)\) \(\chi_{5929}(463,\cdot)\) \(\chi_{5929}(617,\cdot)\) \(\chi_{5929}(694,\cdot)\) \(\chi_{5929}(771,\cdot)\) \(\chi_{5929}(925,\cdot)\) \(\chi_{5929}(1002,\cdot)\) \(\chi_{5929}(1156,\cdot)\) \(\chi_{5929}(1233,\cdot)\) \(\chi_{5929}(1310,\cdot)\) \(\chi_{5929}(1387,\cdot)\) \(\chi_{5929}(1464,\cdot)\) \(\chi_{5929}(1541,\cdot)\) \(\chi_{5929}(1772,\cdot)\) \(\chi_{5929}(1849,\cdot)\) \(\chi_{5929}(1926,\cdot)\) \(\chi_{5929}(2003,\cdot)\) \(\chi_{5929}(2080,\cdot)\) \(\chi_{5929}(2234,\cdot)\) \(\chi_{5929}(2311,\cdot)\) \(\chi_{5929}(2388,\cdot)\) \(\chi_{5929}(2465,\cdot)\) \(\chi_{5929}(2619,\cdot)\) \(\chi_{5929}(2773,\cdot)\) \(\chi_{5929}(2850,\cdot)\) \(\chi_{5929}(2927,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 77 polynomial

Values on generators

\((1816,2059)\) → \((e\left(\frac{3}{7}\right),e\left(\frac{8}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 5929 }(78, a) \) \(1\)\(1\)\(e\left(\frac{67}{77}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{57}{77}\right)\)\(e\left(\frac{19}{77}\right)\)\(e\left(\frac{23}{77}\right)\)\(e\left(\frac{47}{77}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{9}{77}\right)\)\(e\left(\frac{13}{77}\right)\)\(e\left(\frac{46}{77}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5929 }(78,a) \;\) at \(\;a = \) e.g. 2