Properties

Label 7098.2039
Modulus $7098$
Conductor $3549$
Order $156$
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(156))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([78,52,103]))
 
pari: [g,chi] = znchar(Mod(2039,7098))
 

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 \(\chi_{3549}(2039,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7098.eb

\(\chi_{7098}(137,\cdot)\) \(\chi_{7098}(275,\cdot)\) \(\chi_{7098}(305,\cdot)\) \(\chi_{7098}(401,\cdot)\) \(\chi_{7098}(683,\cdot)\) \(\chi_{7098}(821,\cdot)\) \(\chi_{7098}(851,\cdot)\) \(\chi_{7098}(947,\cdot)\) \(\chi_{7098}(1229,\cdot)\) \(\chi_{7098}(1367,\cdot)\) \(\chi_{7098}(1397,\cdot)\) \(\chi_{7098}(1493,\cdot)\) \(\chi_{7098}(1775,\cdot)\) \(\chi_{7098}(1913,\cdot)\) \(\chi_{7098}(1943,\cdot)\) \(\chi_{7098}(2039,\cdot)\) \(\chi_{7098}(2321,\cdot)\) \(\chi_{7098}(2459,\cdot)\) \(\chi_{7098}(2489,\cdot)\) \(\chi_{7098}(2585,\cdot)\) \(\chi_{7098}(2867,\cdot)\) \(\chi_{7098}(3005,\cdot)\) \(\chi_{7098}(3035,\cdot)\) \(\chi_{7098}(3413,\cdot)\) \(\chi_{7098}(3551,\cdot)\) \(\chi_{7098}(3581,\cdot)\) \(\chi_{7098}(3677,\cdot)\) \(\chi_{7098}(3959,\cdot)\) \(\chi_{7098}(4097,\cdot)\) \(\chi_{7098}(4127,\cdot)\) ...

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

\((4733,5071,6931)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{103}{156}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{17}{156}\right)\)\(e\left(\frac{131}{156}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{7}{12}\right)\)\(1\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{71}{78}\right)\)\(e\left(\frac{31}{156}\right)\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{97}{156}\right)\)
value at e.g. 2