Properties

Label 8624.169
Modulus $8624$
Conductor $4312$
Order $70$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,35,40,14]))
 
pari: [g,chi] = znchar(Mod(169,8624))
 

Basic properties

Modulus: \(8624\)
Conductor: \(4312\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4312}(2325,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8624.fi

\(\chi_{8624}(169,\cdot)\) \(\chi_{8624}(729,\cdot)\) \(\chi_{8624}(841,\cdot)\) \(\chi_{8624}(1065,\cdot)\) \(\chi_{8624}(1401,\cdot)\) \(\chi_{8624}(2073,\cdot)\) \(\chi_{8624}(2297,\cdot)\) \(\chi_{8624}(2633,\cdot)\) \(\chi_{8624}(3193,\cdot)\) \(\chi_{8624}(3305,\cdot)\) \(\chi_{8624}(3865,\cdot)\) \(\chi_{8624}(4425,\cdot)\) \(\chi_{8624}(4537,\cdot)\) \(\chi_{8624}(4761,\cdot)\) \(\chi_{8624}(5657,\cdot)\) \(\chi_{8624}(5769,\cdot)\) \(\chi_{8624}(5993,\cdot)\) \(\chi_{8624}(6329,\cdot)\) \(\chi_{8624}(6889,\cdot)\) \(\chi_{8624}(7001,\cdot)\) \(\chi_{8624}(7225,\cdot)\) \(\chi_{8624}(7561,\cdot)\) \(\chi_{8624}(8121,\cdot)\) \(\chi_{8624}(8457,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((5391,6469,7745,3137)\) → \((1,-1,e\left(\frac{4}{7}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 8624 }(169, a) \) \(1\)\(1\)\(e\left(\frac{47}{70}\right)\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{1}{70}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8624 }(169,a) \;\) at \(\;a = \) e.g. 2