Properties

Label 7098.1999
Modulus $7098$
Conductor $1183$
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([0,52,59]))
 
pari: [g,chi] = znchar(Mod(1999,7098))
 

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 \(\chi_{1183}(816,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7098.dm

\(\chi_{7098}(121,\cdot)\) \(\chi_{7098}(667,\cdot)\) \(\chi_{7098}(907,\cdot)\) \(\chi_{7098}(1213,\cdot)\) \(\chi_{7098}(1453,\cdot)\) \(\chi_{7098}(1759,\cdot)\) \(\chi_{7098}(1999,\cdot)\) \(\chi_{7098}(2305,\cdot)\) \(\chi_{7098}(2545,\cdot)\) \(\chi_{7098}(3091,\cdot)\) \(\chi_{7098}(3397,\cdot)\) \(\chi_{7098}(3637,\cdot)\) \(\chi_{7098}(3943,\cdot)\) \(\chi_{7098}(4183,\cdot)\) \(\chi_{7098}(4489,\cdot)\) \(\chi_{7098}(4729,\cdot)\) \(\chi_{7098}(5035,\cdot)\) \(\chi_{7098}(5275,\cdot)\) \(\chi_{7098}(5581,\cdot)\) \(\chi_{7098}(5821,\cdot)\) \(\chi_{7098}(6127,\cdot)\) \(\chi_{7098}(6367,\cdot)\) \(\chi_{7098}(6673,\cdot)\) \(\chi_{7098}(6913,\cdot)\)

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

\((4733,5071,6931)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{59}{78}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{11}{78}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{4}{39}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{23}{78}\right)\)
value at e.g. 2