Properties

Label 1161.17
Modulus $1161$
Conductor $387$
Order $42$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1161, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,38]))
 
pari: [g,chi] = znchar(Mod(17,1161))
 

Basic properties

Modulus: \(1161\)
Conductor: \(387\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{387}(275,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1161.bu

\(\chi_{1161}(17,\cdot)\) \(\chi_{1161}(314,\cdot)\) \(\chi_{1161}(332,\cdot)\) \(\chi_{1161}(440,\cdot)\) \(\chi_{1161}(530,\cdot)\) \(\chi_{1161}(584,\cdot)\) \(\chi_{1161}(611,\cdot)\) \(\chi_{1161}(683,\cdot)\) \(\chi_{1161}(746,\cdot)\) \(\chi_{1161}(926,\cdot)\) \(\chi_{1161}(1115,\cdot)\) \(\chi_{1161}(1142,\cdot)\)

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

\((947,433)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{19}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1161 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1161 }(17,a) \;\) at \(\;a = \) e.g. 2