Properties

Label 2013.188
Modulus $2013$
Conductor $183$
Order $30$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,0,11]))
 
pari: [g,chi] = znchar(Mod(188,2013))
 

Basic properties

Modulus: \(2013\)
Conductor: \(183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{183}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2013.ew

\(\chi_{2013}(188,\cdot)\) \(\chi_{2013}(716,\cdot)\) \(\chi_{2013}(980,\cdot)\) \(\chi_{2013}(1178,\cdot)\) \(\chi_{2013}(1442,\cdot)\) \(\chi_{2013}(1574,\cdot)\) \(\chi_{2013}(1805,\cdot)\) \(\chi_{2013}(1937,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.0.85385482345314369605512628350861103547186790293449369086287.1

Values on generators

\((1343,1465,1222)\) → \((-1,1,e\left(\frac{11}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 2013 }(188, a) \) \(-1\)\(1\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2013 }(188,a) \;\) at \(\;a = \) e.g. 2