Properties

Label 2001.1703
Modulus $2001$
Conductor $87$
Order $28$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,0,17]))
 
pari: [g,chi] = znchar(Mod(1703,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(87\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{87}(50,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2001.bc

\(\chi_{2001}(47,\cdot)\) \(\chi_{2001}(185,\cdot)\) \(\chi_{2001}(392,\cdot)\) \(\chi_{2001}(461,\cdot)\) \(\chi_{2001}(530,\cdot)\) \(\chi_{2001}(599,\cdot)\) \(\chi_{2001}(1013,\cdot)\) \(\chi_{2001}(1220,\cdot)\) \(\chi_{2001}(1634,\cdot)\) \(\chi_{2001}(1703,\cdot)\) \(\chi_{2001}(1772,\cdot)\) \(\chi_{2001}(1841,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: \(\Q(\zeta_{87})^+\)

Values on generators

\((668,1132,553)\) → \((-1,1,e\left(\frac{17}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2001 }(1703, a) \) \(1\)\(1\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2001 }(1703,a) \;\) at \(\;a = \) e.g. 2