Properties

Label 3920.61
Modulus $3920$
Conductor $784$
Order $84$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3920, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,63,0,22]))
 
pari: [g,chi] = znchar(Mod(61,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(784\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{784}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.fo

\(\chi_{3920}(61,\cdot)\) \(\chi_{3920}(101,\cdot)\) \(\chi_{3920}(341,\cdot)\) \(\chi_{3920}(381,\cdot)\) \(\chi_{3920}(621,\cdot)\) \(\chi_{3920}(661,\cdot)\) \(\chi_{3920}(941,\cdot)\) \(\chi_{3920}(1181,\cdot)\) \(\chi_{3920}(1221,\cdot)\) \(\chi_{3920}(1461,\cdot)\) \(\chi_{3920}(1741,\cdot)\) \(\chi_{3920}(1781,\cdot)\) \(\chi_{3920}(2021,\cdot)\) \(\chi_{3920}(2061,\cdot)\) \(\chi_{3920}(2301,\cdot)\) \(\chi_{3920}(2341,\cdot)\) \(\chi_{3920}(2581,\cdot)\) \(\chi_{3920}(2621,\cdot)\) \(\chi_{3920}(2901,\cdot)\) \(\chi_{3920}(3141,\cdot)\) \(\chi_{3920}(3181,\cdot)\) \(\chi_{3920}(3421,\cdot)\) \(\chi_{3920}(3701,\cdot)\) \(\chi_{3920}(3741,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1471,981,3137,3041)\) → \((1,-i,1,e\left(\frac{11}{42}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{5}{6}\right)\)
value at e.g. 2