Properties

Label 2001.620
Modulus $2001$
Conductor $2001$
Order $28$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2001, base_ring=CyclotomicField(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,14,25]))
 
pari: [g,chi] = znchar(Mod(620,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2001.bf

\(\chi_{2001}(68,\cdot)\) \(\chi_{2001}(137,\cdot)\) \(\chi_{2001}(206,\cdot)\) \(\chi_{2001}(275,\cdot)\) \(\chi_{2001}(482,\cdot)\) \(\chi_{2001}(620,\cdot)\) \(\chi_{2001}(827,\cdot)\) \(\chi_{2001}(896,\cdot)\) \(\chi_{2001}(965,\cdot)\) \(\chi_{2001}(1034,\cdot)\) \(\chi_{2001}(1448,\cdot)\) \(\chi_{2001}(1655,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: Number field defined by a degree 28 polynomial

Values on generators

\((668,1132,553)\) → \((-1,-1,e\left(\frac{25}{28}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{4}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2001 }(620,a) \;\) at \(\;a = \) e.g. 2