Properties

Label 2548.209
Modulus $2548$
Conductor $49$
Order $14$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11,0]))
 
pari: [g,chi] = znchar(Mod(209,2548))
 

Basic properties

Modulus: \(2548\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2548.cp

\(\chi_{2548}(209,\cdot)\) \(\chi_{2548}(573,\cdot)\) \(\chi_{2548}(937,\cdot)\) \(\chi_{2548}(1301,\cdot)\) \(\chi_{2548}(2029,\cdot)\) \(\chi_{2548}(2393,\cdot)\)

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: 14.0.1341068619663964900807.1

Values on generators

\((1275,885,197)\) → \((1,e\left(\frac{11}{14}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 2548 }(209, a) \) \(-1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{9}{14}\right)\)\(-1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2548 }(209,a) \;\) at \(\;a = \) e.g. 2