Properties

Label 3920.2287
Modulus $3920$
Conductor $980$
Order $84$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: 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([42,0,21,82]))
 
pari: [g,chi] = znchar(Mod(2287,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(980\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{980}(327,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.fn

\(\chi_{3920}(47,\cdot)\) \(\chi_{3920}(143,\cdot)\) \(\chi_{3920}(367,\cdot)\) \(\chi_{3920}(383,\cdot)\) \(\chi_{3920}(703,\cdot)\) \(\chi_{3920}(927,\cdot)\) \(\chi_{3920}(943,\cdot)\) \(\chi_{3920}(1167,\cdot)\) \(\chi_{3920}(1263,\cdot)\) \(\chi_{3920}(1487,\cdot)\) \(\chi_{3920}(1503,\cdot)\) \(\chi_{3920}(1727,\cdot)\) \(\chi_{3920}(1823,\cdot)\) \(\chi_{3920}(2047,\cdot)\) \(\chi_{3920}(2063,\cdot)\) \(\chi_{3920}(2287,\cdot)\) \(\chi_{3920}(2607,\cdot)\) \(\chi_{3920}(2623,\cdot)\) \(\chi_{3920}(2847,\cdot)\) \(\chi_{3920}(2943,\cdot)\) \(\chi_{3920}(3183,\cdot)\) \(\chi_{3920}(3407,\cdot)\) \(\chi_{3920}(3503,\cdot)\) \(\chi_{3920}(3727,\cdot)\)

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

\((1471,981,3137,3041)\) → \((-1,1,i,e\left(\frac{41}{42}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{55}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3920 }(2287,a) \;\) at \(\;a = \) e.g. 2