Properties

Label 8400.67
Modulus $8400$
Conductor $2800$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8400, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,45,0,39,40]))
 
pari: [g,chi] = znchar(Mod(67,8400))
 

Basic properties

Modulus: \(8400\)
Conductor: \(2800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2800}(67,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8400.kx

\(\chi_{8400}(67,\cdot)\) \(\chi_{8400}(1003,\cdot)\) \(\chi_{8400}(1747,\cdot)\) \(\chi_{8400}(2683,\cdot)\) \(\chi_{8400}(2923,\cdot)\) \(\chi_{8400}(3187,\cdot)\) \(\chi_{8400}(3427,\cdot)\) \(\chi_{8400}(4363,\cdot)\) \(\chi_{8400}(4603,\cdot)\) \(\chi_{8400}(4867,\cdot)\) \(\chi_{8400}(6283,\cdot)\) \(\chi_{8400}(6547,\cdot)\) \(\chi_{8400}(6787,\cdot)\) \(\chi_{8400}(7723,\cdot)\) \(\chi_{8400}(7963,\cdot)\) \(\chi_{8400}(8227,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((3151,2101,2801,5377,3601)\) → \((-1,-i,1,e\left(\frac{13}{20}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8400 }(67, a) \) \(1\)\(1\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8400 }(67,a) \;\) at \(\;a = \) e.g. 2