Properties

Label 2600.167
Modulus $2600$
Conductor $1300$
Order $60$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2600, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,0,39,35]))
 
pari: [g,chi] = znchar(Mod(167,2600))
 

Basic properties

Modulus: \(2600\)
Conductor: \(1300\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1300}(167,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2600.fw

\(\chi_{2600}(167,\cdot)\) \(\chi_{2600}(223,\cdot)\) \(\chi_{2600}(383,\cdot)\) \(\chi_{2600}(527,\cdot)\) \(\chi_{2600}(687,\cdot)\) \(\chi_{2600}(903,\cdot)\) \(\chi_{2600}(1047,\cdot)\) \(\chi_{2600}(1263,\cdot)\) \(\chi_{2600}(1423,\cdot)\) \(\chi_{2600}(1567,\cdot)\) \(\chi_{2600}(1727,\cdot)\) \(\chi_{2600}(1783,\cdot)\) \(\chi_{2600}(2087,\cdot)\) \(\chi_{2600}(2247,\cdot)\) \(\chi_{2600}(2303,\cdot)\) \(\chi_{2600}(2463,\cdot)\)

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

\((1951,1301,1977,1601)\) → \((-1,1,e\left(\frac{13}{20}\right),e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 2600 }(167, a) \) \(-1\)\(1\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{19}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2600 }(167,a) \;\) at \(\;a = \) e.g. 2