Properties

Label 1127.55
Modulus $1127$
Conductor $1127$
Order $154$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1127, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([99,70]))
 
pari: [g,chi] = znchar(Mod(55,1127))
 

Basic properties

Modulus: \(1127\)
Conductor: \(1127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
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 1127.z

\(\chi_{1127}(6,\cdot)\) \(\chi_{1127}(13,\cdot)\) \(\chi_{1127}(27,\cdot)\) \(\chi_{1127}(41,\cdot)\) \(\chi_{1127}(55,\cdot)\) \(\chi_{1127}(62,\cdot)\) \(\chi_{1127}(104,\cdot)\) \(\chi_{1127}(118,\cdot)\) \(\chi_{1127}(167,\cdot)\) \(\chi_{1127}(174,\cdot)\) \(\chi_{1127}(188,\cdot)\) \(\chi_{1127}(202,\cdot)\) \(\chi_{1127}(209,\cdot)\) \(\chi_{1127}(216,\cdot)\) \(\chi_{1127}(223,\cdot)\) \(\chi_{1127}(265,\cdot)\) \(\chi_{1127}(279,\cdot)\) \(\chi_{1127}(307,\cdot)\) \(\chi_{1127}(328,\cdot)\) \(\chi_{1127}(335,\cdot)\) \(\chi_{1127}(349,\cdot)\) \(\chi_{1127}(363,\cdot)\) \(\chi_{1127}(370,\cdot)\) \(\chi_{1127}(377,\cdot)\) \(\chi_{1127}(384,\cdot)\) \(\chi_{1127}(426,\cdot)\) \(\chi_{1127}(468,\cdot)\) \(\chi_{1127}(496,\cdot)\) \(\chi_{1127}(510,\cdot)\) \(\chi_{1127}(524,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((346,442)\) → \((e\left(\frac{9}{14}\right),e\left(\frac{5}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1127 }(55, a) \) \(-1\)\(1\)\(e\left(\frac{48}{77}\right)\)\(e\left(\frac{141}{154}\right)\)\(e\left(\frac{19}{77}\right)\)\(e\left(\frac{15}{154}\right)\)\(e\left(\frac{83}{154}\right)\)\(e\left(\frac{67}{77}\right)\)\(e\left(\frac{64}{77}\right)\)\(e\left(\frac{111}{154}\right)\)\(e\left(\frac{62}{77}\right)\)\(e\left(\frac{25}{154}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1127 }(55,a) \;\) at \(\;a = \) e.g. 2