Properties

Label 1148.15
Modulus $1148$
Conductor $164$
Order $40$
Real no
Primitive no
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(1148, base_ring=CyclotomicField(40))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([20,0,37]))
 
pari: [g,chi] = znchar(Mod(15,1148))
 

Basic properties

Modulus: \(1148\)
Conductor: \(164\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{164}(15,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1148.ca

\(\chi_{1148}(15,\cdot)\) \(\chi_{1148}(71,\cdot)\) \(\chi_{1148}(99,\cdot)\) \(\chi_{1148}(183,\cdot)\) \(\chi_{1148}(211,\cdot)\) \(\chi_{1148}(239,\cdot)\) \(\chi_{1148}(463,\cdot)\) \(\chi_{1148}(603,\cdot)\) \(\chi_{1148}(827,\cdot)\) \(\chi_{1148}(855,\cdot)\) \(\chi_{1148}(883,\cdot)\) \(\chi_{1148}(967,\cdot)\) \(\chi_{1148}(995,\cdot)\) \(\chi_{1148}(1051,\cdot)\) \(\chi_{1148}(1079,\cdot)\) \(\chi_{1148}(1135,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{40})\)
Fixed field: \(\Q(\zeta_{164})^+\)

Values on generators

\((575,493,785)\) → \((-1,1,e\left(\frac{37}{40}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{20}\right)\)\(-i\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{21}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)
value at e.g. 2