Properties

Label 4410.1349
Modulus $4410$
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(4410, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,21,17]))
 
pari: [g,chi] = znchar(Mod(1349,4410))
 

Basic properties

Modulus: \(4410\)
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 \(\chi_{735}(614,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.dc

\(\chi_{4410}(89,\cdot)\) \(\chi_{4410}(269,\cdot)\) \(\chi_{4410}(719,\cdot)\) \(\chi_{4410}(899,\cdot)\) \(\chi_{4410}(1349,\cdot)\) \(\chi_{4410}(1529,\cdot)\) \(\chi_{4410}(2159,\cdot)\) \(\chi_{4410}(2609,\cdot)\) \(\chi_{4410}(2789,\cdot)\) \(\chi_{4410}(3239,\cdot)\) \(\chi_{4410}(3419,\cdot)\) \(\chi_{4410}(3869,\cdot)\)

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

\((3431,2647,1081)\) → \((-1,-1,e\left(\frac{17}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4410 }(1349, a) \) \(1\)\(1\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{13}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4410 }(1349,a) \;\) at \(\;a = \) e.g. 2