Properties

Label 6272.703
Modulus $6272$
Conductor $392$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6272, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,21,25]))
 
pari: [g,chi] = znchar(Mod(703,6272))
 

Basic properties

Modulus: \(6272\)
Conductor: \(392\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{392}(115,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6272.ce

\(\chi_{6272}(703,\cdot)\) \(\chi_{6272}(831,\cdot)\) \(\chi_{6272}(1727,\cdot)\) \(\chi_{6272}(2495,\cdot)\) \(\chi_{6272}(2623,\cdot)\) \(\chi_{6272}(3391,\cdot)\) \(\chi_{6272}(3519,\cdot)\) \(\chi_{6272}(4287,\cdot)\) \(\chi_{6272}(4415,\cdot)\) \(\chi_{6272}(5183,\cdot)\) \(\chi_{6272}(6079,\cdot)\) \(\chi_{6272}(6207,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.42.1090030896264192289800449659845679818091197961133776603876122561317234873686091104256.1

Values on generators

\((4607,3333,4609)\) → \((-1,-1,e\left(\frac{25}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 6272 }(703, a) \) \(1\)\(1\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{11}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6272 }(703,a) \;\) at \(\;a = \) e.g. 2