Properties

Label 3072.73
Modulus $3072$
Conductor $512$
Order $128$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3072, base_ring=CyclotomicField(128))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,91,0]))
 
pari: [g,chi] = znchar(Mod(73,3072))
 

Basic properties

Modulus: \(3072\)
Conductor: \(512\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(128\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{512}(157,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3072.bd

\(\chi_{3072}(25,\cdot)\) \(\chi_{3072}(73,\cdot)\) \(\chi_{3072}(121,\cdot)\) \(\chi_{3072}(169,\cdot)\) \(\chi_{3072}(217,\cdot)\) \(\chi_{3072}(265,\cdot)\) \(\chi_{3072}(313,\cdot)\) \(\chi_{3072}(361,\cdot)\) \(\chi_{3072}(409,\cdot)\) \(\chi_{3072}(457,\cdot)\) \(\chi_{3072}(505,\cdot)\) \(\chi_{3072}(553,\cdot)\) \(\chi_{3072}(601,\cdot)\) \(\chi_{3072}(649,\cdot)\) \(\chi_{3072}(697,\cdot)\) \(\chi_{3072}(745,\cdot)\) \(\chi_{3072}(793,\cdot)\) \(\chi_{3072}(841,\cdot)\) \(\chi_{3072}(889,\cdot)\) \(\chi_{3072}(937,\cdot)\) \(\chi_{3072}(985,\cdot)\) \(\chi_{3072}(1033,\cdot)\) \(\chi_{3072}(1081,\cdot)\) \(\chi_{3072}(1129,\cdot)\) \(\chi_{3072}(1177,\cdot)\) \(\chi_{3072}(1225,\cdot)\) \(\chi_{3072}(1273,\cdot)\) \(\chi_{3072}(1321,\cdot)\) \(\chi_{3072}(1369,\cdot)\) \(\chi_{3072}(1417,\cdot)\) ...

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: $\Q(\zeta_{128})$
Fixed field: Number field defined by a degree 128 polynomial (not computed)

Values on generators

\((2047,2053,1025)\) → \((1,e\left(\frac{91}{128}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 3072 }(73, a) \) \(1\)\(1\)\(e\left(\frac{91}{128}\right)\)\(e\left(\frac{39}{64}\right)\)\(e\left(\frac{55}{128}\right)\)\(e\left(\frac{117}{128}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{45}{128}\right)\)\(e\left(\frac{61}{64}\right)\)\(e\left(\frac{27}{64}\right)\)\(e\left(\frac{57}{128}\right)\)\(e\left(\frac{11}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3072 }(73,a) \;\) at \(\;a = \) e.g. 2