Properties

Label 6034.57
Modulus $6034$
Conductor $431$
Order $215$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, base_ring=CyclotomicField(430))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,24]))
 
pari: [g,chi] = znchar(Mod(57,6034))
 

Basic properties

Modulus: \(6034\)
Conductor: \(431\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(215\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{431}(57,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6034.v

\(\chi_{6034}(15,\cdot)\) \(\chi_{6034}(29,\cdot)\) \(\chi_{6034}(57,\cdot)\) \(\chi_{6034}(99,\cdot)\) \(\chi_{6034}(169,\cdot)\) \(\chi_{6034}(183,\cdot)\) \(\chi_{6034}(197,\cdot)\) \(\chi_{6034}(225,\cdot)\) \(\chi_{6034}(253,\cdot)\) \(\chi_{6034}(295,\cdot)\) \(\chi_{6034}(379,\cdot)\) \(\chi_{6034}(477,\cdot)\) \(\chi_{6034}(491,\cdot)\) \(\chi_{6034}(519,\cdot)\) \(\chi_{6034}(631,\cdot)\) \(\chi_{6034}(659,\cdot)\) \(\chi_{6034}(673,\cdot)\) \(\chi_{6034}(701,\cdot)\) \(\chi_{6034}(757,\cdot)\) \(\chi_{6034}(785,\cdot)\) \(\chi_{6034}(799,\cdot)\) \(\chi_{6034}(827,\cdot)\) \(\chi_{6034}(841,\cdot)\) \(\chi_{6034}(855,\cdot)\) \(\chi_{6034}(911,\cdot)\) \(\chi_{6034}(953,\cdot)\) \(\chi_{6034}(981,\cdot)\) \(\chi_{6034}(1009,\cdot)\) \(\chi_{6034}(1121,\cdot)\) \(\chi_{6034}(1135,\cdot)\) ...

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: $\Q(\zeta_{215})$
Fixed field: Number field defined by a degree 215 polynomial (not computed)

Values on generators

\((1725,869)\) → \((1,e\left(\frac{12}{215}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 6034 }(57, a) \) \(1\)\(1\)\(e\left(\frac{27}{43}\right)\)\(e\left(\frac{19}{215}\right)\)\(e\left(\frac{11}{43}\right)\)\(e\left(\frac{71}{215}\right)\)\(e\left(\frac{69}{215}\right)\)\(e\left(\frac{154}{215}\right)\)\(e\left(\frac{157}{215}\right)\)\(e\left(\frac{153}{215}\right)\)\(e\left(\frac{42}{215}\right)\)\(e\left(\frac{38}{215}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6034 }(57,a) \;\) at \(\;a = \) e.g. 2