Properties

Label 1176.149
Modulus $1176$
Conductor $1176$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,21,26]))
 
pari: [g,chi] = znchar(Mod(149,1176))
 

Basic properties

Modulus: \(1176\)
Conductor: \(1176\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 1176.ck

\(\chi_{1176}(53,\cdot)\) \(\chi_{1176}(149,\cdot)\) \(\chi_{1176}(221,\cdot)\) \(\chi_{1176}(317,\cdot)\) \(\chi_{1176}(389,\cdot)\) \(\chi_{1176}(485,\cdot)\) \(\chi_{1176}(653,\cdot)\) \(\chi_{1176}(725,\cdot)\) \(\chi_{1176}(821,\cdot)\) \(\chi_{1176}(893,\cdot)\) \(\chi_{1176}(989,\cdot)\) \(\chi_{1176}(1061,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((295,589,785,1081)\) → \((1,-1,-1,e\left(\frac{13}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1176 }(149, a) \) \(-1\)\(1\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1176 }(149,a) \;\) at \(\;a = \) e.g. 2