Properties

Label 3549.88
Modulus $3549$
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(3549, base_ring=CyclotomicField(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,52,53]))
 
pari: [g,chi] = znchar(Mod(88,3549))
 

Basic properties

Modulus: \(3549\)
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 \(\chi_{1183}(88,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.de

\(\chi_{3549}(88,\cdot)\) \(\chi_{3549}(121,\cdot)\) \(\chi_{3549}(394,\cdot)\) \(\chi_{3549}(634,\cdot)\) \(\chi_{3549}(667,\cdot)\) \(\chi_{3549}(907,\cdot)\) \(\chi_{3549}(940,\cdot)\) \(\chi_{3549}(1180,\cdot)\) \(\chi_{3549}(1213,\cdot)\) \(\chi_{3549}(1453,\cdot)\) \(\chi_{3549}(1486,\cdot)\) \(\chi_{3549}(1726,\cdot)\) \(\chi_{3549}(1759,\cdot)\) \(\chi_{3549}(1999,\cdot)\) \(\chi_{3549}(2032,\cdot)\) \(\chi_{3549}(2272,\cdot)\) \(\chi_{3549}(2305,\cdot)\) \(\chi_{3549}(2545,\cdot)\) \(\chi_{3549}(2578,\cdot)\) \(\chi_{3549}(2818,\cdot)\) \(\chi_{3549}(3091,\cdot)\) \(\chi_{3549}(3124,\cdot)\) \(\chi_{3549}(3364,\cdot)\) \(\chi_{3549}(3397,\cdot)\)

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

\((1184,1522,3382)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{53}{78}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{35}{78}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{34}{39}\right)\)\(-1\)\(e\left(\frac{37}{78}\right)\)
value at e.g. 2