Properties

Label 7098.317
Modulus $7098$
Conductor $3549$
Order $156$
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(156))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([78,52,153]))
 
pari: [g,chi] = znchar(Mod(317,7098))
 

Basic properties

Modulus: \(7098\)
Conductor: \(3549\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3549}(317,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7098.ej

\(\chi_{7098}(317,\cdot)\) \(\chi_{7098}(359,\cdot)\) \(\chi_{7098}(473,\cdot)\) \(\chi_{7098}(515,\cdot)\) \(\chi_{7098}(863,\cdot)\) \(\chi_{7098}(905,\cdot)\) \(\chi_{7098}(1019,\cdot)\) \(\chi_{7098}(1061,\cdot)\) \(\chi_{7098}(1409,\cdot)\) \(\chi_{7098}(1565,\cdot)\) \(\chi_{7098}(1607,\cdot)\) \(\chi_{7098}(1955,\cdot)\) \(\chi_{7098}(1997,\cdot)\) \(\chi_{7098}(2111,\cdot)\) \(\chi_{7098}(2153,\cdot)\) \(\chi_{7098}(2501,\cdot)\) \(\chi_{7098}(2543,\cdot)\) \(\chi_{7098}(2657,\cdot)\) \(\chi_{7098}(2699,\cdot)\) \(\chi_{7098}(3047,\cdot)\) \(\chi_{7098}(3089,\cdot)\) \(\chi_{7098}(3203,\cdot)\) \(\chi_{7098}(3245,\cdot)\) \(\chi_{7098}(3593,\cdot)\) \(\chi_{7098}(3635,\cdot)\) \(\chi_{7098}(3749,\cdot)\) \(\chi_{7098}(3791,\cdot)\) \(\chi_{7098}(4139,\cdot)\) \(\chi_{7098}(4181,\cdot)\) \(\chi_{7098}(4337,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((4733,5071,6931)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{51}{52}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{155}{156}\right)\)\(e\left(\frac{133}{156}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{77}{78}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{145}{156}\right)\)\(e\left(\frac{119}{156}\right)\)\(e\left(\frac{45}{52}\right)\)
value at e.g. 2