Properties

Label 7098.419
Modulus $7098$
Conductor $3549$
Order $78$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(7098, base_ring=CyclotomicField(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([39,39,14]))
 
pari: [g,chi] = znchar(Mod(419,7098))
 

Basic properties

Modulus: \(7098\)
Conductor: \(3549\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3549}(419,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7098.cz

\(\chi_{7098}(419,\cdot)\) \(\chi_{7098}(503,\cdot)\) \(\chi_{7098}(965,\cdot)\) \(\chi_{7098}(1049,\cdot)\) \(\chi_{7098}(1511,\cdot)\) \(\chi_{7098}(1595,\cdot)\) \(\chi_{7098}(2057,\cdot)\) \(\chi_{7098}(2141,\cdot)\) \(\chi_{7098}(2603,\cdot)\) \(\chi_{7098}(2687,\cdot)\) \(\chi_{7098}(3149,\cdot)\) \(\chi_{7098}(3779,\cdot)\) \(\chi_{7098}(4241,\cdot)\) \(\chi_{7098}(4325,\cdot)\) \(\chi_{7098}(4787,\cdot)\) \(\chi_{7098}(4871,\cdot)\) \(\chi_{7098}(5333,\cdot)\) \(\chi_{7098}(5417,\cdot)\) \(\chi_{7098}(5879,\cdot)\) \(\chi_{7098}(5963,\cdot)\) \(\chi_{7098}(6425,\cdot)\) \(\chi_{7098}(6509,\cdot)\) \(\chi_{7098}(6971,\cdot)\) \(\chi_{7098}(7055,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((4733,5071,6931)\) → \((-1,-1,e\left(\frac{7}{39}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{77}{78}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{10}{39}\right)\)
value at e.g. 2