Properties

Label 7098.205
Modulus $7098$
Conductor $1183$
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([0,26,47]))
 
pari: [g,chi] = znchar(Mod(205,7098))
 

Basic properties

Modulus: \(7098\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1183}(205,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7098.dd

\(\chi_{7098}(205,\cdot)\) \(\chi_{7098}(277,\cdot)\) \(\chi_{7098}(751,\cdot)\) \(\chi_{7098}(1297,\cdot)\) \(\chi_{7098}(1369,\cdot)\) \(\chi_{7098}(1843,\cdot)\) \(\chi_{7098}(1915,\cdot)\) \(\chi_{7098}(2461,\cdot)\) \(\chi_{7098}(2935,\cdot)\) \(\chi_{7098}(3007,\cdot)\) \(\chi_{7098}(3481,\cdot)\) \(\chi_{7098}(3553,\cdot)\) \(\chi_{7098}(4027,\cdot)\) \(\chi_{7098}(4099,\cdot)\) \(\chi_{7098}(4573,\cdot)\) \(\chi_{7098}(4645,\cdot)\) \(\chi_{7098}(5119,\cdot)\) \(\chi_{7098}(5191,\cdot)\) \(\chi_{7098}(5665,\cdot)\) \(\chi_{7098}(5737,\cdot)\) \(\chi_{7098}(6211,\cdot)\) \(\chi_{7098}(6283,\cdot)\) \(\chi_{7098}(6757,\cdot)\) \(\chi_{7098}(6829,\cdot)\)

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

\((4733,5071,6931)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{47}{78}\right))\)

Values

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