Properties

Label 4033.3072
Modulus $4033$
Conductor $109$
Order $54$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4033, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,41]))
 
pari: [g,chi] = znchar(Mod(3072,4033))
 

Basic properties

Modulus: \(4033\)
Conductor: \(109\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{109}(20,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4033.hj

\(\chi_{4033}(519,\cdot)\) \(\chi_{4033}(741,\cdot)\) \(\chi_{4033}(1259,\cdot)\) \(\chi_{4033}(1555,\cdot)\) \(\chi_{4033}(1666,\cdot)\) \(\chi_{4033}(2036,\cdot)\) \(\chi_{4033}(2591,\cdot)\) \(\chi_{4033}(2628,\cdot)\) \(\chi_{4033}(2813,\cdot)\) \(\chi_{4033}(3072,\cdot)\) \(\chi_{4033}(3146,\cdot)\) \(\chi_{4033}(3331,\cdot)\) \(\chi_{4033}(3479,\cdot)\) \(\chi_{4033}(3516,\cdot)\) \(\chi_{4033}(3590,\cdot)\) \(\chi_{4033}(3701,\cdot)\) \(\chi_{4033}(3812,\cdot)\) \(\chi_{4033}(3960,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((1963,2295)\) → \((1,e\left(\frac{41}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(3072, a) \) \(1\)\(1\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{13}{27}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{53}{54}\right)\)\(e\left(\frac{1}{54}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(3072,a) \;\) at \(\;a = \) e.g. 2