Properties

Label 5070.107
Modulus $5070$
Conductor $2535$
Order $156$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5070, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,39,100]))
 
pari: [g,chi] = znchar(Mod(107,5070))
 

Basic properties

Modulus: \(5070\)
Conductor: \(2535\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2535}(107,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5070.ct

\(\chi_{5070}(107,\cdot)\) \(\chi_{5070}(113,\cdot)\) \(\chi_{5070}(263,\cdot)\) \(\chi_{5070}(347,\cdot)\) \(\chi_{5070}(497,\cdot)\) \(\chi_{5070}(503,\cdot)\) \(\chi_{5070}(737,\cdot)\) \(\chi_{5070}(887,\cdot)\) \(\chi_{5070}(893,\cdot)\) \(\chi_{5070}(1043,\cdot)\) \(\chi_{5070}(1127,\cdot)\) \(\chi_{5070}(1277,\cdot)\) \(\chi_{5070}(1283,\cdot)\) \(\chi_{5070}(1433,\cdot)\) \(\chi_{5070}(1517,\cdot)\) \(\chi_{5070}(1673,\cdot)\) \(\chi_{5070}(1823,\cdot)\) \(\chi_{5070}(1907,\cdot)\) \(\chi_{5070}(2057,\cdot)\) \(\chi_{5070}(2063,\cdot)\) \(\chi_{5070}(2213,\cdot)\) \(\chi_{5070}(2297,\cdot)\) \(\chi_{5070}(2447,\cdot)\) \(\chi_{5070}(2453,\cdot)\) \(\chi_{5070}(2603,\cdot)\) \(\chi_{5070}(2687,\cdot)\) \(\chi_{5070}(2837,\cdot)\) \(\chi_{5070}(2843,\cdot)\) \(\chi_{5070}(2993,\cdot)\) \(\chi_{5070}(3077,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((1691,4057,1861)\) → \((-1,i,e\left(\frac{25}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 5070 }(107, a) \) \(1\)\(1\)\(e\left(\frac{131}{156}\right)\)\(e\left(\frac{41}{78}\right)\)\(e\left(\frac{53}{156}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{7}{156}\right)\)\(e\left(\frac{77}{78}\right)\)\(e\left(\frac{149}{156}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5070 }(107,a) \;\) at \(\;a = \) e.g. 2