Properties

Label 4004.25
Modulus $4004$
Conductor $1001$
Order $30$
Real no
Primitive no
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([0,20,24,15]))
 
pari: [g,chi] = znchar(Mod(25,4004))
 

Basic properties

Modulus: \(4004\)
Conductor: \(1001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1001}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4004.hg

\(\chi_{4004}(25,\cdot)\) \(\chi_{4004}(389,\cdot)\) \(\chi_{4004}(597,\cdot)\) \(\chi_{4004}(753,\cdot)\) \(\chi_{4004}(961,\cdot)\) \(\chi_{4004}(1325,\cdot)\) \(\chi_{4004}(2209,\cdot)\) \(\chi_{4004}(2781,\cdot)\)

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

\((2003,3433,365,925)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)\(29\)
\( \chi_{ 4004 }(25, a) \) \(1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4004 }(25,a) \;\) at \(\;a = \) e.g. 2