Properties

Label 1157.1004
Modulus $1157$
Conductor $1157$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1157, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,39]))
 
pari: [g,chi] = znchar(Mod(1004,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 1157.bp

\(\chi_{1157}(22,\cdot)\) \(\chi_{1157}(81,\cdot)\) \(\chi_{1157}(87,\cdot)\) \(\chi_{1157}(100,\cdot)\) \(\chi_{1157}(133,\cdot)\) \(\chi_{1157}(139,\cdot)\) \(\chi_{1157}(146,\cdot)\) \(\chi_{1157}(263,\cdot)\) \(\chi_{1157}(289,\cdot)\) \(\chi_{1157}(354,\cdot)\) \(\chi_{1157}(367,\cdot)\) \(\chi_{1157}(406,\cdot)\) \(\chi_{1157}(607,\cdot)\) \(\chi_{1157}(737,\cdot)\) \(\chi_{1157}(874,\cdot)\) \(\chi_{1157}(971,\cdot)\) \(\chi_{1157}(1004,\cdot)\) \(\chi_{1157}(1023,\cdot)\) \(\chi_{1157}(1036,\cdot)\) \(\chi_{1157}(1153,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((535,92)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{13}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(1004, a) \) \(1\)\(1\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{32}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(1004,a) \;\) at \(\;a = \) e.g. 2