Properties

Label 9240.1597
Modulus $9240$
Conductor $440$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9240, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,10,0,5,0,2]))
 
pari: [g,chi] = znchar(Mod(1597,9240))
 

Basic properties

Modulus: \(9240\)
Conductor: \(440\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{440}(277,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9240.kx

\(\chi_{9240}(1597,\cdot)\) \(\chi_{9240}(1933,\cdot)\) \(\chi_{9240}(2437,\cdot)\) \(\chi_{9240}(4957,\cdot)\) \(\chi_{9240}(5293,\cdot)\) \(\chi_{9240}(6133,\cdot)\) \(\chi_{9240}(7477,\cdot)\) \(\chi_{9240}(8653,\cdot)\)

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: 20.20.182187370528513441169408000000000000000.1

Values on generators

\((2311,4621,6161,3697,5281,2521)\) → \((1,-1,1,i,1,e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 9240 }(1597, a) \) \(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(-i\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(-i\)\(e\left(\frac{1}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9240 }(1597,a) \;\) at \(\;a = \) e.g. 2