Properties

Label 4033.1685
Modulus $4033$
Conductor $4033$
Order $108$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4033, base_ring=CyclotomicField(108))
 
M = H._module
 
chi = DirichletCharacter(H, M([75,101]))
 
pari: [g,chi] = znchar(Mod(1685,4033))
 

Basic properties

Modulus: \(4033\)
Conductor: \(4033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4033.jz

\(\chi_{4033}(69,\cdot)\) \(\chi_{4033}(167,\cdot)\) \(\chi_{4033}(316,\cdot)\) \(\chi_{4033}(449,\cdot)\) \(\chi_{4033}(483,\cdot)\) \(\chi_{4033}(503,\cdot)\) \(\chi_{4033}(531,\cdot)\) \(\chi_{4033}(648,\cdot)\) \(\chi_{4033}(753,\cdot)\) \(\chi_{4033}(819,\cdot)\) \(\chi_{4033}(890,\cdot)\) \(\chi_{4033}(1169,\cdot)\) \(\chi_{4033}(1236,\cdot)\) \(\chi_{4033}(1238,\cdot)\) \(\chi_{4033}(1393,\cdot)\) \(\chi_{4033}(1685,\cdot)\) \(\chi_{4033}(1700,\cdot)\) \(\chi_{4033}(1905,\cdot)\) \(\chi_{4033}(2128,\cdot)\) \(\chi_{4033}(2333,\cdot)\) \(\chi_{4033}(2348,\cdot)\) \(\chi_{4033}(2640,\cdot)\) \(\chi_{4033}(2795,\cdot)\) \(\chi_{4033}(2797,\cdot)\) \(\chi_{4033}(2864,\cdot)\) \(\chi_{4033}(3143,\cdot)\) \(\chi_{4033}(3214,\cdot)\) \(\chi_{4033}(3280,\cdot)\) \(\chi_{4033}(3385,\cdot)\) \(\chi_{4033}(3502,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\((1963,2295)\) → \((e\left(\frac{25}{36}\right),e\left(\frac{101}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(1685, a) \) \(1\)\(1\)\(1\)\(e\left(\frac{37}{54}\right)\)\(1\)\(e\left(\frac{5}{108}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{17}{27}\right)\)\(1\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{5}{108}\right)\)\(e\left(\frac{49}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(1685,a) \;\) at \(\;a = \) e.g. 2