Properties

Label 4056.71
Modulus $4056$
Conductor $2028$
Order $156$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,0,78,137]))
 
pari: [g,chi] = znchar(Mod(71,4056))
 

Basic properties

Modulus: \(4056\)
Conductor: \(2028\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2028}(71,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4056.dq

\(\chi_{4056}(71,\cdot)\) \(\chi_{4056}(119,\cdot)\) \(\chi_{4056}(167,\cdot)\) \(\chi_{4056}(215,\cdot)\) \(\chi_{4056}(383,\cdot)\) \(\chi_{4056}(431,\cdot)\) \(\chi_{4056}(479,\cdot)\) \(\chi_{4056}(527,\cdot)\) \(\chi_{4056}(743,\cdot)\) \(\chi_{4056}(791,\cdot)\) \(\chi_{4056}(839,\cdot)\) \(\chi_{4056}(1007,\cdot)\) \(\chi_{4056}(1055,\cdot)\) \(\chi_{4056}(1151,\cdot)\) \(\chi_{4056}(1319,\cdot)\) \(\chi_{4056}(1367,\cdot)\) \(\chi_{4056}(1415,\cdot)\) \(\chi_{4056}(1463,\cdot)\) \(\chi_{4056}(1631,\cdot)\) \(\chi_{4056}(1679,\cdot)\) \(\chi_{4056}(1727,\cdot)\) \(\chi_{4056}(1775,\cdot)\) \(\chi_{4056}(1943,\cdot)\) \(\chi_{4056}(1991,\cdot)\) \(\chi_{4056}(2039,\cdot)\) \(\chi_{4056}(2087,\cdot)\) \(\chi_{4056}(2255,\cdot)\) \(\chi_{4056}(2303,\cdot)\) \(\chi_{4056}(2351,\cdot)\) \(\chi_{4056}(2399,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((1015,2029,2705,3889)\) → \((-1,1,-1,e\left(\frac{137}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 4056 }(71, a) \) \(-1\)\(1\)\(e\left(\frac{21}{52}\right)\)\(e\left(\frac{73}{156}\right)\)\(e\left(\frac{71}{156}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{49}{78}\right)\)\(e\left(\frac{49}{52}\right)\)\(e\left(\frac{34}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4056 }(71,a) \;\) at \(\;a = \) e.g. 2