Properties

Label 1157.525
Modulus $1157$
Conductor $1157$
Order $44$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1157, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,23]))
 
pari: [g,chi] = znchar(Mod(525,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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 1157.bj

\(\chi_{1157}(5,\cdot)\) \(\chi_{1157}(21,\cdot)\) \(\chi_{1157}(109,\cdot)\) \(\chi_{1157}(125,\cdot)\) \(\chi_{1157}(138,\cdot)\) \(\chi_{1157}(346,\cdot)\) \(\chi_{1157}(398,\cdot)\) \(\chi_{1157}(463,\cdot)\) \(\chi_{1157}(525,\cdot)\) \(\chi_{1157}(551,\cdot)\) \(\chi_{1157}(606,\cdot)\) \(\chi_{1157}(632,\cdot)\) \(\chi_{1157}(694,\cdot)\) \(\chi_{1157}(759,\cdot)\) \(\chi_{1157}(811,\cdot)\) \(\chi_{1157}(1019,\cdot)\) \(\chi_{1157}(1032,\cdot)\) \(\chi_{1157}(1048,\cdot)\) \(\chi_{1157}(1136,\cdot)\) \(\chi_{1157}(1152,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((535,92)\) → \((-i,e\left(\frac{23}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(525, a) \) \(-1\)\(1\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(525,a) \;\) at \(\;a = \) e.g. 2