Properties

Label 1148.29
Modulus $1148$
Conductor $41$
Order $40$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

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([0,0,7]))
 
pari: [g,chi] = znchar(Mod(29,1148))
 

Basic properties

Modulus: \(1148\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1148.cd

\(\chi_{1148}(29,\cdot)\) \(\chi_{1148}(253,\cdot)\) \(\chi_{1148}(281,\cdot)\) \(\chi_{1148}(309,\cdot)\) \(\chi_{1148}(393,\cdot)\) \(\chi_{1148}(421,\cdot)\) \(\chi_{1148}(477,\cdot)\) \(\chi_{1148}(505,\cdot)\) \(\chi_{1148}(561,\cdot)\) \(\chi_{1148}(589,\cdot)\) \(\chi_{1148}(645,\cdot)\) \(\chi_{1148}(673,\cdot)\) \(\chi_{1148}(757,\cdot)\) \(\chi_{1148}(785,\cdot)\) \(\chi_{1148}(813,\cdot)\) \(\chi_{1148}(1037,\cdot)\)

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: \(\Q(\zeta_{41})\)

Values on generators

\((575,493,785)\) → \((1,1,e\left(\frac{7}{40}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{17}{20}\right)\)\(i\)\(e\left(\frac{21}{40}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{10}\right)\)
value at e.g. 2