Properties

Label 3822.145
Modulus $3822$
Conductor $637$
Order $84$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,10,7]))
 
pari: [g,chi] = znchar(Mod(145,3822))
 

Basic properties

Modulus: \(3822\)
Conductor: \(637\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{637}(145,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3822.eb

\(\chi_{3822}(145,\cdot)\) \(\chi_{3822}(241,\cdot)\) \(\chi_{3822}(271,\cdot)\) \(\chi_{3822}(409,\cdot)\) \(\chi_{3822}(691,\cdot)\) \(\chi_{3822}(787,\cdot)\) \(\chi_{3822}(817,\cdot)\) \(\chi_{3822}(955,\cdot)\) \(\chi_{3822}(1237,\cdot)\) \(\chi_{3822}(1333,\cdot)\) \(\chi_{3822}(1363,\cdot)\) \(\chi_{3822}(1879,\cdot)\) \(\chi_{3822}(1909,\cdot)\) \(\chi_{3822}(2047,\cdot)\) \(\chi_{3822}(2329,\cdot)\) \(\chi_{3822}(2425,\cdot)\) \(\chi_{3822}(2455,\cdot)\) \(\chi_{3822}(2593,\cdot)\) \(\chi_{3822}(2875,\cdot)\) \(\chi_{3822}(3001,\cdot)\) \(\chi_{3822}(3139,\cdot)\) \(\chi_{3822}(3421,\cdot)\) \(\chi_{3822}(3517,\cdot)\) \(\chi_{3822}(3685,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((2549,3433,1471)\) → \((1,e\left(\frac{5}{42}\right),e\left(\frac{1}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3822 }(145, a) \) \(1\)\(1\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{73}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3822 }(145,a) \;\) at \(\;a = \) e.g. 2