Properties

Label 1521.10
Modulus $1521$
Conductor $169$
Order $78$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5]))
 
pari: [g,chi] = znchar(Mod(10,1521))
 

Basic properties

Modulus: \(1521\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(10,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1521.bt

\(\chi_{1521}(10,\cdot)\) \(\chi_{1521}(82,\cdot)\) \(\chi_{1521}(127,\cdot)\) \(\chi_{1521}(199,\cdot)\) \(\chi_{1521}(244,\cdot)\) \(\chi_{1521}(433,\cdot)\) \(\chi_{1521}(478,\cdot)\) \(\chi_{1521}(550,\cdot)\) \(\chi_{1521}(595,\cdot)\) \(\chi_{1521}(667,\cdot)\) \(\chi_{1521}(712,\cdot)\) \(\chi_{1521}(784,\cdot)\) \(\chi_{1521}(829,\cdot)\) \(\chi_{1521}(901,\cdot)\) \(\chi_{1521}(946,\cdot)\) \(\chi_{1521}(1018,\cdot)\) \(\chi_{1521}(1063,\cdot)\) \(\chi_{1521}(1135,\cdot)\) \(\chi_{1521}(1180,\cdot)\) \(\chi_{1521}(1252,\cdot)\) \(\chi_{1521}(1297,\cdot)\) \(\chi_{1521}(1369,\cdot)\) \(\chi_{1521}(1414,\cdot)\) \(\chi_{1521}(1486,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((677,847)\) → \((1,e\left(\frac{5}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 1521 }(10, a) \) \(1\)\(1\)\(e\left(\frac{5}{78}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{67}{78}\right)\)\(e\left(\frac{5}{26}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{14}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1521 }(10,a) \;\) at \(\;a = \) e.g. 2