Properties

Label 4004.75
Modulus $4004$
Conductor $4004$
Order $30$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,25,18,25]))
 
pari: [g,chi] = znchar(Mod(75,4004))
 

Basic properties

Modulus: \(4004\)
Conductor: \(4004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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 4004.hm

\(\chi_{4004}(75,\cdot)\) \(\chi_{4004}(927,\cdot)\) \(\chi_{4004}(1291,\cdot)\) \(\chi_{4004}(1655,\cdot)\) \(\chi_{4004}(1895,\cdot)\) \(\chi_{4004}(2259,\cdot)\) \(\chi_{4004}(2623,\cdot)\) \(\chi_{4004}(3111,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((2003,3433,365,925)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{3}{5}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)\(29\)
\( \chi_{ 4004 }(75, a) \) \(1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{19}{30}\right)\)\(-1\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{8}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4004 }(75,a) \;\) at \(\;a = \) e.g. 2