Properties

Label 4400.19
Modulus $4400$
Conductor $4400$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,15,18,6]))
 
pari: [g,chi] = znchar(Mod(19,4400))
 

Basic properties

Modulus: \(4400\)
Conductor: \(4400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 4400.gr

\(\chi_{4400}(19,\cdot)\) \(\chi_{4400}(259,\cdot)\) \(\chi_{4400}(579,\cdot)\) \(\chi_{4400}(739,\cdot)\) \(\chi_{4400}(2219,\cdot)\) \(\chi_{4400}(2459,\cdot)\) \(\chi_{4400}(2779,\cdot)\) \(\chi_{4400}(2939,\cdot)\)

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: 20.20.116599917138248602348421120000000000000000000000000000000000.3

Values on generators

\((2751,3301,177,1201)\) → \((-1,-i,e\left(\frac{9}{10}\right),e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4400 }(19, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4400 }(19,a) \;\) at \(\;a = \) e.g. 2