Properties

Label 2850.139
Modulus $2850$
Conductor $475$
Order $90$
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(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,27,70]))
 
pari: [g,chi] = znchar(Mod(139,2850))
 

Basic properties

Modulus: \(2850\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(139,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2850.cp

\(\chi_{2850}(139,\cdot)\) \(\chi_{2850}(169,\cdot)\) \(\chi_{2850}(289,\cdot)\) \(\chi_{2850}(529,\cdot)\) \(\chi_{2850}(709,\cdot)\) \(\chi_{2850}(739,\cdot)\) \(\chi_{2850}(769,\cdot)\) \(\chi_{2850}(859,\cdot)\) \(\chi_{2850}(1069,\cdot)\) \(\chi_{2850}(1279,\cdot)\) \(\chi_{2850}(1309,\cdot)\) \(\chi_{2850}(1339,\cdot)\) \(\chi_{2850}(1429,\cdot)\) \(\chi_{2850}(1639,\cdot)\) \(\chi_{2850}(1669,\cdot)\) \(\chi_{2850}(1879,\cdot)\) \(\chi_{2850}(1909,\cdot)\) \(\chi_{2850}(2209,\cdot)\) \(\chi_{2850}(2239,\cdot)\) \(\chi_{2850}(2419,\cdot)\) \(\chi_{2850}(2479,\cdot)\) \(\chi_{2850}(2569,\cdot)\) \(\chi_{2850}(2779,\cdot)\) \(\chi_{2850}(2809,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((1901,1027,1351)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{7}{9}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{53}{90}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{77}{90}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{17}{18}\right)\)
value at e.g. 2