Properties

Label 1157.25
Modulus $1157$
Conductor $1157$
Order $22$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1157)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,13]))
 
pari: [g,chi] = znchar(Mod(25,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
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.bb

\(\chi_{1157}(25,\cdot)\) \(\chi_{1157}(259,\cdot)\) \(\chi_{1157}(311,\cdot)\) \(\chi_{1157}(324,\cdot)\) \(\chi_{1157}(441,\cdot)\) \(\chi_{1157}(467,\cdot)\) \(\chi_{1157}(532,\cdot)\) \(\chi_{1157}(545,\cdot)\) \(\chi_{1157}(584,\cdot)\) \(\chi_{1157}(1052,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{22}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial