Properties

Label 3549.43
Modulus $3549$
Conductor $169$
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([0,0,61]))
 
pari: [g,chi] = znchar(Mod(43,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.dm

\(\chi_{3549}(43,\cdot)\) \(\chi_{3549}(127,\cdot)\) \(\chi_{3549}(400,\cdot)\) \(\chi_{3549}(589,\cdot)\) \(\chi_{3549}(673,\cdot)\) \(\chi_{3549}(862,\cdot)\) \(\chi_{3549}(946,\cdot)\) \(\chi_{3549}(1135,\cdot)\) \(\chi_{3549}(1219,\cdot)\) \(\chi_{3549}(1408,\cdot)\) \(\chi_{3549}(1492,\cdot)\) \(\chi_{3549}(1681,\cdot)\) \(\chi_{3549}(1765,\cdot)\) \(\chi_{3549}(1954,\cdot)\) \(\chi_{3549}(2038,\cdot)\) \(\chi_{3549}(2227,\cdot)\) \(\chi_{3549}(2311,\cdot)\) \(\chi_{3549}(2500,\cdot)\) \(\chi_{3549}(2584,\cdot)\) \(\chi_{3549}(2773,\cdot)\) \(\chi_{3549}(2857,\cdot)\) \(\chi_{3549}(3046,\cdot)\) \(\chi_{3549}(3130,\cdot)\) \(\chi_{3549}(3319,\cdot)\)

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

\((1184,1522,3382)\) → \((1,1,e\left(\frac{61}{78}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{7}{39}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{47}{78}\right)\)
value at e.g. 2