Properties

Label 1984.37
Modulus $1984$
Conductor $1984$
Order $48$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1984, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,27,40]))
 
pari: [g,chi] = znchar(Mod(37,1984))
 

Basic properties

Modulus: \(1984\)
Conductor: \(1984\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1984.cl

\(\chi_{1984}(37,\cdot)\) \(\chi_{1984}(181,\cdot)\) \(\chi_{1984}(285,\cdot)\) \(\chi_{1984}(429,\cdot)\) \(\chi_{1984}(533,\cdot)\) \(\chi_{1984}(677,\cdot)\) \(\chi_{1984}(781,\cdot)\) \(\chi_{1984}(925,\cdot)\) \(\chi_{1984}(1029,\cdot)\) \(\chi_{1984}(1173,\cdot)\) \(\chi_{1984}(1277,\cdot)\) \(\chi_{1984}(1421,\cdot)\) \(\chi_{1984}(1525,\cdot)\) \(\chi_{1984}(1669,\cdot)\) \(\chi_{1984}(1773,\cdot)\) \(\chi_{1984}(1917,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((63,1861,65)\) → \((1,e\left(\frac{9}{16}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1984 }(37, a) \) \(-1\)\(1\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{29}{48}\right)\)\(-i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{48}\right)\)\(e\left(\frac{23}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1984 }(37,a) \;\) at \(\;a = \) e.g. 2