Properties

Label 4002.25
Modulus $4002$
Conductor $667$
Order $77$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,14,88]))
 
pari: [g,chi] = znchar(Mod(25,4002))
 

Basic properties

Modulus: \(4002\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(77\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4002.bk

\(\chi_{4002}(25,\cdot)\) \(\chi_{4002}(49,\cdot)\) \(\chi_{4002}(169,\cdot)\) \(\chi_{4002}(223,\cdot)\) \(\chi_{4002}(397,\cdot)\) \(\chi_{4002}(487,\cdot)\) \(\chi_{4002}(547,\cdot)\) \(\chi_{4002}(625,\cdot)\) \(\chi_{4002}(703,\cdot)\) \(\chi_{4002}(721,\cdot)\) \(\chi_{4002}(745,\cdot)\) \(\chi_{4002}(877,\cdot)\) \(\chi_{4002}(1039,\cdot)\) \(\chi_{4002}(1051,\cdot)\) \(\chi_{4002}(1093,\cdot)\) \(\chi_{4002}(1225,\cdot)\) \(\chi_{4002}(1267,\cdot)\) \(\chi_{4002}(1531,\cdot)\) \(\chi_{4002}(1573,\cdot)\) \(\chi_{4002}(1591,\cdot)\) \(\chi_{4002}(1669,\cdot)\) \(\chi_{4002}(1705,\cdot)\) \(\chi_{4002}(1789,\cdot)\) \(\chi_{4002}(1843,\cdot)\) \(\chi_{4002}(1879,\cdot)\) \(\chi_{4002}(1921,\cdot)\) \(\chi_{4002}(1963,\cdot)\) \(\chi_{4002}(2017,\cdot)\) \(\chi_{4002}(2053,\cdot)\) \(\chi_{4002}(2083,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 77 polynomial

Values on generators

\((2669,3133,553)\) → \((1,e\left(\frac{1}{11}\right),e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 4002 }(25, a) \) \(1\)\(1\)\(e\left(\frac{51}{77}\right)\)\(e\left(\frac{45}{77}\right)\)\(e\left(\frac{8}{77}\right)\)\(e\left(\frac{43}{77}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{39}{77}\right)\)\(e\left(\frac{25}{77}\right)\)\(e\left(\frac{9}{77}\right)\)\(e\left(\frac{19}{77}\right)\)\(e\left(\frac{48}{77}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4002 }(25,a) \;\) at \(\;a = \) e.g. 2