Properties

Label 1148.415
Modulus $1148$
Conductor $1148$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,20,33]))
 
pari: [g,chi] = znchar(Mod(415,1148))
 

Basic properties

Modulus: \(1148\)
Conductor: \(1148\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 1148.ce

\(\chi_{1148}(39,\cdot)\) \(\chi_{1148}(207,\cdot)\) \(\chi_{1148}(415,\cdot)\) \(\chi_{1148}(431,\cdot)\) \(\chi_{1148}(443,\cdot)\) \(\chi_{1148}(459,\cdot)\) \(\chi_{1148}(471,\cdot)\) \(\chi_{1148}(487,\cdot)\) \(\chi_{1148}(695,\cdot)\) \(\chi_{1148}(863,\cdot)\) \(\chi_{1148}(907,\cdot)\) \(\chi_{1148}(935,\cdot)\) \(\chi_{1148}(963,\cdot)\) \(\chi_{1148}(1087,\cdot)\) \(\chi_{1148}(1115,\cdot)\) \(\chi_{1148}(1143,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((575,493,785)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{11}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 1148 }(415, a) \) \(-1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{8}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1148 }(415,a) \;\) at \(\;a = \) e.g. 2