Properties

Label 2850.77
Modulus $2850$
Conductor $75$
Order $20$
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(2850, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,1,0]))
 
pari: [g,chi] = znchar(Mod(77,2850))
 

Basic properties

Modulus: \(2850\)
Conductor: \(75\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{75}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2850.br

\(\chi_{2850}(77,\cdot)\) \(\chi_{2850}(533,\cdot)\) \(\chi_{2850}(647,\cdot)\) \(\chi_{2850}(1103,\cdot)\) \(\chi_{2850}(1217,\cdot)\) \(\chi_{2850}(1673,\cdot)\) \(\chi_{2850}(1787,\cdot)\) \(\chi_{2850}(2813,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{75})^+\)

Values on generators

\((1901,1027,1351)\) → \((-1,e\left(\frac{1}{20}\right),1)\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(i\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(-i\)
value at e.g. 2