Properties

Conductor 25
Order 20
Real No
Primitive No
Parity Odd
Orbit Label 4000.bo

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(4000)
 
sage: chi = H[2657]
 
pari: [g,chi] = znchar(Mod(2657,4000))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 25
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 20
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Odd
Orbit label = 4000.bo
Orbit index = 41

Galois orbit

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

\(\chi_{4000}(257,\cdot)\) \(\chi_{4000}(993,\cdot)\) \(\chi_{4000}(1793,\cdot)\) \(\chi_{4000}(1857,\cdot)\) \(\chi_{4000}(2593,\cdot)\) \(\chi_{4000}(2657,\cdot)\) \(\chi_{4000}(3393,\cdot)\) \(\chi_{4000}(3457,\cdot)\)

Inducing primitive character

\(\chi_{25}(2,\cdot)\)

Values on generators

\((2751,2501,1377)\) → \((1,1,e\left(\frac{1}{20}\right))\)

Values

-1137911131719212327
\(-1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(i\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{20}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{20})\)