Properties

Label 7350.299
Modulus $7350$
Conductor $735$
Order $42$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7350, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,21,29]))
 
pari: [g,chi] = znchar(Mod(299,7350))
 

Basic properties

Modulus: \(7350\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{735}(299,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7350.ch

\(\chi_{7350}(299,\cdot)\) \(\chi_{7350}(899,\cdot)\) \(\chi_{7350}(1349,\cdot)\) \(\chi_{7350}(1949,\cdot)\) \(\chi_{7350}(2399,\cdot)\) \(\chi_{7350}(2999,\cdot)\) \(\chi_{7350}(4499,\cdot)\) \(\chi_{7350}(5099,\cdot)\) \(\chi_{7350}(5549,\cdot)\) \(\chi_{7350}(6149,\cdot)\) \(\chi_{7350}(6599,\cdot)\) \(\chi_{7350}(7199,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.42.589475176645907082922286550311127085690444572711075874815443834048428072939395904541015625.1

Values on generators

\((4901,1177,2551)\) → \((-1,-1,e\left(\frac{29}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 7350 }(299, a) \) \(1\)\(1\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{9}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7350 }(299,a) \;\) at \(\;a = \) e.g. 2