Properties

Label 4706.421
Modulus $4706$
Conductor $2353$
Order $20$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4706, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,4]))
 
pari: [g,chi] = znchar(Mod(421,4706))
 

Basic properties

Modulus: \(4706\)
Conductor: \(2353\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2353}(421,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4706.cu

\(\chi_{4706}(125,\cdot)\) \(\chi_{4706}(135,\cdot)\) \(\chi_{4706}(421,\cdot)\) \(\chi_{4706}(1945,\cdot)\) \(\chi_{4706}(2033,\cdot)\) \(\chi_{4706}(2231,\cdot)\) \(\chi_{4706}(3021,\cdot)\) \(\chi_{4706}(3843,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((2173,183)\) → \((-i,e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 4706 }(421, a) \) \(-1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(-1\)\(-i\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4706 }(421,a) \;\) at \(\;a = \) e.g. 2