Properties

Label 3344.217
Modulus $3344$
Conductor $1672$
Order $30$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,9,5]))
 
pari: [g,chi] = znchar(Mod(217,3344))
 

Basic properties

Modulus: \(3344\)
Conductor: \(1672\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1672}(1053,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3344.dv

\(\chi_{3344}(217,\cdot)\) \(\chi_{3344}(601,\cdot)\) \(\chi_{3344}(1129,\cdot)\) \(\chi_{3344}(1513,\cdot)\) \(\chi_{3344}(1817,\cdot)\) \(\chi_{3344}(2041,\cdot)\) \(\chi_{3344}(2345,\cdot)\) \(\chi_{3344}(3033,\cdot)\)

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: Number field defined by a degree 30 polynomial

Values on generators

\((2927,837,2433,705)\) → \((1,-1,e\left(\frac{3}{10}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 3344 }(217, a) \) \(1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3344 }(217,a) \;\) at \(\;a = \) e.g. 2