Properties

Label 2013.200
Modulus $2013$
Conductor $2013$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,6,47]))
 
pari: [g,chi] = znchar(Mod(200,2013))
 

Basic properties

Modulus: \(2013\)
Conductor: \(2013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 2013.fz

\(\chi_{2013}(200,\cdot)\) \(\chi_{2013}(299,\cdot)\) \(\chi_{2013}(359,\cdot)\) \(\chi_{2013}(458,\cdot)\) \(\chi_{2013}(734,\cdot)\) \(\chi_{2013}(767,\cdot)\) \(\chi_{2013}(860,\cdot)\) \(\chi_{2013}(959,\cdot)\) \(\chi_{2013}(1238,\cdot)\) \(\chi_{2013}(1250,\cdot)\) \(\chi_{2013}(1271,\cdot)\) \(\chi_{2013}(1349,\cdot)\) \(\chi_{2013}(1535,\cdot)\) \(\chi_{2013}(1568,\cdot)\) \(\chi_{2013}(1856,\cdot)\) \(\chi_{2013}(1889,\cdot)\)

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

\((1343,1465,1222)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{47}{60}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 2013 }(200, a) \) \(-1\)\(1\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2013 }(200,a) \;\) at \(\;a = \) e.g. 2