Properties

Label 7350.1051
Modulus $7350$
Conductor $49$
Order $7$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7350, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,8]))
 
pari: [g,chi] = znchar(Mod(1051,7350))
 

Basic properties

Modulus: \(7350\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(7\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(22,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7350.v

\(\chi_{7350}(1051,\cdot)\) \(\chi_{7350}(2101,\cdot)\) \(\chi_{7350}(3151,\cdot)\) \(\chi_{7350}(4201,\cdot)\) \(\chi_{7350}(5251,\cdot)\) \(\chi_{7350}(6301,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 7.7.13841287201.1

Values on generators

\((4901,1177,2551)\) → \((1,1,e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 7350 }(1051, a) \) \(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(1\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7350 }(1051,a) \;\) at \(\;a = \) e.g. 2