Properties

Label 7098.311
Modulus $7098$
Conductor $3549$
Order $78$
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(7098, base_ring=CyclotomicField(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([39,13,69]))
 
pari: [g,chi] = znchar(Mod(311,7098))
 

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 \(\chi_{3549}(311,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7098.dj

\(\chi_{7098}(311,\cdot)\) \(\chi_{7098}(467,\cdot)\) \(\chi_{7098}(857,\cdot)\) \(\chi_{7098}(1403,\cdot)\) \(\chi_{7098}(1559,\cdot)\) \(\chi_{7098}(1949,\cdot)\) \(\chi_{7098}(2105,\cdot)\) \(\chi_{7098}(2495,\cdot)\) \(\chi_{7098}(2651,\cdot)\) \(\chi_{7098}(3197,\cdot)\) \(\chi_{7098}(3587,\cdot)\) \(\chi_{7098}(3743,\cdot)\) \(\chi_{7098}(4133,\cdot)\) \(\chi_{7098}(4289,\cdot)\) \(\chi_{7098}(4679,\cdot)\) \(\chi_{7098}(4835,\cdot)\) \(\chi_{7098}(5225,\cdot)\) \(\chi_{7098}(5381,\cdot)\) \(\chi_{7098}(5771,\cdot)\) \(\chi_{7098}(5927,\cdot)\) \(\chi_{7098}(6317,\cdot)\) \(\chi_{7098}(6473,\cdot)\) \(\chi_{7098}(6863,\cdot)\) \(\chi_{7098}(7019,\cdot)\)

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

\((4733,5071,6931)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{23}{26}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{23}{78}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{23}{39}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{29}{39}\right)\)\(e\left(\frac{71}{78}\right)\)\(e\left(\frac{5}{26}\right)\)
value at e.g. 2