Properties

Label 4034.899
Modulus $4034$
Conductor $2017$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4034, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([4]))
 
pari: [g,chi] = znchar(Mod(899,4034))
 

Basic properties

Modulus: \(4034\)
Conductor: \(2017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2017}(899,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4034.m

\(\chi_{4034}(899,\cdot)\) \(\chi_{4034}(1039,\cdot)\) \(\chi_{4034}(1401,\cdot)\) \(\chi_{4034}(1761,\cdot)\) \(\chi_{4034}(1785,\cdot)\) \(\chi_{4034}(1963,\cdot)\) \(\chi_{4034}(2277,\cdot)\) \(\chi_{4034}(2387,\cdot)\) \(\chi_{4034}(2443,\cdot)\) \(\chi_{4034}(3009,\cdot)\) \(\chi_{4034}(3399,\cdot)\) \(\chi_{4034}(3859,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\(5\) → \(e\left(\frac{2}{21}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 4034 }(899, a) \) \(1\)\(1\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{13}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4034 }(899,a) \;\) at \(\;a = \) e.g. 2