Properties

Label 9408.169
Modulus $9408$
Conductor $1568$
Order $56$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9408, base_ring=CyclotomicField(56))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,49,0,32]))
 
pari: [g,chi] = znchar(Mod(169,9408))
 

Basic properties

Modulus: \(9408\)
Conductor: \(1568\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(56\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1568}(365,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9408.ex

\(\chi_{9408}(169,\cdot)\) \(\chi_{9408}(505,\cdot)\) \(\chi_{9408}(841,\cdot)\) \(\chi_{9408}(1513,\cdot)\) \(\chi_{9408}(1849,\cdot)\) \(\chi_{9408}(2185,\cdot)\) \(\chi_{9408}(2521,\cdot)\) \(\chi_{9408}(2857,\cdot)\) \(\chi_{9408}(3193,\cdot)\) \(\chi_{9408}(3865,\cdot)\) \(\chi_{9408}(4201,\cdot)\) \(\chi_{9408}(4537,\cdot)\) \(\chi_{9408}(4873,\cdot)\) \(\chi_{9408}(5209,\cdot)\) \(\chi_{9408}(5545,\cdot)\) \(\chi_{9408}(6217,\cdot)\) \(\chi_{9408}(6553,\cdot)\) \(\chi_{9408}(6889,\cdot)\) \(\chi_{9408}(7225,\cdot)\) \(\chi_{9408}(7561,\cdot)\) \(\chi_{9408}(7897,\cdot)\) \(\chi_{9408}(8569,\cdot)\) \(\chi_{9408}(8905,\cdot)\) \(\chi_{9408}(9241,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((1471,6469,3137,4609)\) → \((1,e\left(\frac{7}{8}\right),1,e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 9408 }(169, a) \) \(1\)\(1\)\(e\left(\frac{25}{56}\right)\)\(e\left(\frac{13}{56}\right)\)\(e\left(\frac{55}{56}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{51}{56}\right)\)\(1\)\(e\left(\frac{9}{56}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9408 }(169,a) \;\) at \(\;a = \) e.g. 2