Properties

Label 2850.83
Modulus $2850$
Conductor $1425$
Order $60$
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(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([30,9,20]))
 
pari: [g,chi] = znchar(Mod(83,2850))
 

Basic properties

Modulus: \(2850\)
Conductor: \(1425\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1425}(83,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2850.ci

\(\chi_{2850}(83,\cdot)\) \(\chi_{2850}(197,\cdot)\) \(\chi_{2850}(353,\cdot)\) \(\chi_{2850}(467,\cdot)\) \(\chi_{2850}(653,\cdot)\) \(\chi_{2850}(767,\cdot)\) \(\chi_{2850}(923,\cdot)\) \(\chi_{2850}(1037,\cdot)\) \(\chi_{2850}(1223,\cdot)\) \(\chi_{2850}(1337,\cdot)\) \(\chi_{2850}(2063,\cdot)\) \(\chi_{2850}(2177,\cdot)\) \(\chi_{2850}(2363,\cdot)\) \(\chi_{2850}(2477,\cdot)\) \(\chi_{2850}(2633,\cdot)\) \(\chi_{2850}(2747,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1901,1027,1351)\) → \((-1,e\left(\frac{3}{20}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(-i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{7}{12}\right)\)
value at e.g. 2