Properties

Label 9408.85
Modulus $9408$
Conductor $3136$
Order $112$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9408, base_ring=CyclotomicField(112))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,91,0,32]))
 
pari: [g,chi] = znchar(Mod(85,9408))
 

Basic properties

Modulus: \(9408\)
Conductor: \(3136\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(112\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3136}(85,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9408.fh

\(\chi_{9408}(85,\cdot)\) \(\chi_{9408}(253,\cdot)\) \(\chi_{9408}(421,\cdot)\) \(\chi_{9408}(757,\cdot)\) \(\chi_{9408}(925,\cdot)\) \(\chi_{9408}(1093,\cdot)\) \(\chi_{9408}(1261,\cdot)\) \(\chi_{9408}(1429,\cdot)\) \(\chi_{9408}(1597,\cdot)\) \(\chi_{9408}(1933,\cdot)\) \(\chi_{9408}(2101,\cdot)\) \(\chi_{9408}(2269,\cdot)\) \(\chi_{9408}(2437,\cdot)\) \(\chi_{9408}(2605,\cdot)\) \(\chi_{9408}(2773,\cdot)\) \(\chi_{9408}(3109,\cdot)\) \(\chi_{9408}(3277,\cdot)\) \(\chi_{9408}(3445,\cdot)\) \(\chi_{9408}(3613,\cdot)\) \(\chi_{9408}(3781,\cdot)\) \(\chi_{9408}(3949,\cdot)\) \(\chi_{9408}(4285,\cdot)\) \(\chi_{9408}(4453,\cdot)\) \(\chi_{9408}(4621,\cdot)\) \(\chi_{9408}(4789,\cdot)\) \(\chi_{9408}(4957,\cdot)\) \(\chi_{9408}(5125,\cdot)\) \(\chi_{9408}(5461,\cdot)\) \(\chi_{9408}(5629,\cdot)\) \(\chi_{9408}(5797,\cdot)\) ...

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: $\Q(\zeta_{112})$
Fixed field: Number field defined by a degree 112 polynomial (not computed)

Values on generators

\((1471,6469,3137,4609)\) → \((1,e\left(\frac{13}{16}\right),1,e\left(\frac{2}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 9408 }(85, a) \) \(1\)\(1\)\(e\left(\frac{11}{112}\right)\)\(e\left(\frac{55}{112}\right)\)\(e\left(\frac{69}{112}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{13}{56}\right)\)\(e\left(\frac{11}{56}\right)\)\(e\left(\frac{9}{112}\right)\)\(-1\)\(e\left(\frac{51}{112}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9408 }(85,a) \;\) at \(\;a = \) e.g. 2