Properties

Label 8008.15
Modulus $8008$
Conductor $572$
Order $60$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,0,0,12,5]))
 
pari: [g,chi] = znchar(Mod(15,8008))
 

Basic properties

Modulus: \(8008\)
Conductor: \(572\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{572}(15,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8008.qo

\(\chi_{8008}(15,\cdot)\) \(\chi_{8008}(71,\cdot)\) \(\chi_{8008}(631,\cdot)\) \(\chi_{8008}(687,\cdot)\) \(\chi_{8008}(1527,\cdot)\) \(\chi_{8008}(2143,\cdot)\) \(\chi_{8008}(3655,\cdot)\) \(\chi_{8008}(3711,\cdot)\) \(\chi_{8008}(4271,\cdot)\) \(\chi_{8008}(4327,\cdot)\) \(\chi_{8008}(4383,\cdot)\) \(\chi_{8008}(4999,\cdot)\) \(\chi_{8008}(5839,\cdot)\) \(\chi_{8008}(6455,\cdot)\) \(\chi_{8008}(7351,\cdot)\) \(\chi_{8008}(7967,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((6007,4005,3433,4369,4929)\) → \((-1,1,1,e\left(\frac{1}{5}\right),e\left(\frac{1}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)\(29\)
\( \chi_{ 8008 }(15, a) \) \(1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8008 }(15,a) \;\) at \(\;a = \) e.g. 2