Properties

Label 2704.1301
Modulus $2704$
Conductor $2704$
Order $52$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, base_ring=CyclotomicField(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,13,12]))
 
pari: [g,chi] = znchar(Mod(1301,2704))
 

Basic properties

Modulus: \(2704\)
Conductor: \(2704\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2704.cd

\(\chi_{2704}(53,\cdot)\) \(\chi_{2704}(157,\cdot)\) \(\chi_{2704}(261,\cdot)\) \(\chi_{2704}(365,\cdot)\) \(\chi_{2704}(469,\cdot)\) \(\chi_{2704}(573,\cdot)\) \(\chi_{2704}(781,\cdot)\) \(\chi_{2704}(885,\cdot)\) \(\chi_{2704}(989,\cdot)\) \(\chi_{2704}(1093,\cdot)\) \(\chi_{2704}(1197,\cdot)\) \(\chi_{2704}(1301,\cdot)\) \(\chi_{2704}(1405,\cdot)\) \(\chi_{2704}(1509,\cdot)\) \(\chi_{2704}(1613,\cdot)\) \(\chi_{2704}(1717,\cdot)\) \(\chi_{2704}(1821,\cdot)\) \(\chi_{2704}(1925,\cdot)\) \(\chi_{2704}(2133,\cdot)\) \(\chi_{2704}(2237,\cdot)\) \(\chi_{2704}(2341,\cdot)\) \(\chi_{2704}(2445,\cdot)\) \(\chi_{2704}(2549,\cdot)\) \(\chi_{2704}(2653,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((2367,677,1185)\) → \((1,i,e\left(\frac{3}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 2704 }(1301, a) \) \(1\)\(1\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{17}{52}\right)\)\(e\left(\frac{5}{26}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{1}{52}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(-i\)\(e\left(\frac{29}{52}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2704 }(1301,a) \;\) at \(\;a = \) e.g. 2