Properties

Label 1407.125
Modulus $1407$
Conductor $1407$
Order $22$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1407, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,11,15]))
 
pari: [g,chi] = znchar(Mod(125,1407))
 

Basic properties

Modulus: \(1407\)
Conductor: \(1407\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1407.bq

\(\chi_{1407}(125,\cdot)\) \(\chi_{1407}(209,\cdot)\) \(\chi_{1407}(377,\cdot)\) \(\chi_{1407}(608,\cdot)\) \(\chi_{1407}(713,\cdot)\) \(\chi_{1407}(923,\cdot)\) \(\chi_{1407}(965,\cdot)\) \(\chi_{1407}(1259,\cdot)\) \(\chi_{1407}(1343,\cdot)\) \(\chi_{1407}(1385,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\((470,1207,337)\) → \((-1,-1,e\left(\frac{15}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1407 }(125, a) \) \(-1\)\(1\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{7}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1407 }(125,a) \;\) at \(\;a = \) e.g. 2