Properties

Label 1157.628
Modulus $1157$
Conductor $1157$
Order $132$
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(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,105]))
 
pari: [g,chi] = znchar(Mod(628,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
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.bu

\(\chi_{1157}(10,\cdot)\) \(\chi_{1157}(17,\cdot)\) \(\chi_{1157}(36,\cdot)\) \(\chi_{1157}(49,\cdot)\) \(\chi_{1157}(69,\cdot)\) \(\chi_{1157}(160,\cdot)\) \(\chi_{1157}(173,\cdot)\) \(\chi_{1157}(199,\cdot)\) \(\chi_{1157}(218,\cdot)\) \(\chi_{1157}(225,\cdot)\) \(\chi_{1157}(231,\cdot)\) \(\chi_{1157}(257,\cdot)\) \(\chi_{1157}(277,\cdot)\) \(\chi_{1157}(303,\cdot)\) \(\chi_{1157}(309,\cdot)\) \(\chi_{1157}(316,\cdot)\) \(\chi_{1157}(335,\cdot)\) \(\chi_{1157}(361,\cdot)\) \(\chi_{1157}(374,\cdot)\) \(\chi_{1157}(465,\cdot)\) \(\chi_{1157}(485,\cdot)\) \(\chi_{1157}(498,\cdot)\) \(\chi_{1157}(517,\cdot)\) \(\chi_{1157}(524,\cdot)\) \(\chi_{1157}(543,\cdot)\) \(\chi_{1157}(576,\cdot)\) \(\chi_{1157}(602,\cdot)\) \(\chi_{1157}(628,\cdot)\) \(\chi_{1157}(641,\cdot)\) \(\chi_{1157}(732,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((535,92)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{35}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(628, a) \) \(1\)\(1\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{61}{132}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{47}{132}\right)\)\(e\left(\frac{35}{132}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{65}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(628,a) \;\) at \(\;a = \) e.g. 2