Properties

Label 7098.251
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([39,39,43]))
 
pari: [g,chi] = znchar(Mod(251,7098))
 

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 \(\chi_{3549}(251,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7098.dz

\(\chi_{7098}(251,\cdot)\) \(\chi_{7098}(335,\cdot)\) \(\chi_{7098}(797,\cdot)\) \(\chi_{7098}(881,\cdot)\) \(\chi_{7098}(1343,\cdot)\) \(\chi_{7098}(1427,\cdot)\) \(\chi_{7098}(1889,\cdot)\) \(\chi_{7098}(1973,\cdot)\) \(\chi_{7098}(2435,\cdot)\) \(\chi_{7098}(2519,\cdot)\) \(\chi_{7098}(2981,\cdot)\) \(\chi_{7098}(3611,\cdot)\) \(\chi_{7098}(4073,\cdot)\) \(\chi_{7098}(4157,\cdot)\) \(\chi_{7098}(4619,\cdot)\) \(\chi_{7098}(4703,\cdot)\) \(\chi_{7098}(5165,\cdot)\) \(\chi_{7098}(5249,\cdot)\) \(\chi_{7098}(5711,\cdot)\) \(\chi_{7098}(5795,\cdot)\) \(\chi_{7098}(6257,\cdot)\) \(\chi_{7098}(6341,\cdot)\) \(\chi_{7098}(6803,\cdot)\) \(\chi_{7098}(6887,\cdot)\)

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

\((4733,5071,6931)\) → \((-1,-1,e\left(\frac{43}{78}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{19}{78}\right)\)\(e\left(\frac{67}{78}\right)\)
value at e.g. 2