Properties

Label 85600.47937
Modulus $85600$
Conductor $25$
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(85600, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,9,0]))
 
pari: [g,chi] = znchar(Mod(47937,85600))
 

Basic properties

Modulus: \(85600\)
Conductor: \(25\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{25}(12,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 85600.dd

\(\chi_{85600}(10273,\cdot)\) \(\chi_{85600}(13697,\cdot)\) \(\chi_{85600}(30817,\cdot)\) \(\chi_{85600}(44513,\cdot)\) \(\chi_{85600}(47937,\cdot)\) \(\chi_{85600}(61633,\cdot)\) \(\chi_{85600}(78753,\cdot)\) \(\chi_{85600}(82177,\cdot)\)

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

\((26751,32101,82177,16801)\) → \((1,1,e\left(\frac{9}{20}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 85600 }(47937, a) \) \(-1\)\(1\)\(e\left(\frac{3}{20}\right)\)\(i\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{9}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 85600 }(47937,a) \;\) at \(\;a = \) e.g. 2