Properties

Label 3724.2825
Modulus $3724$
Conductor $931$
Order $126$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3724, base_ring=CyclotomicField(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,12,35]))
 
pari: [g,chi] = znchar(Mod(2825,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{931}(32,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.ed

\(\chi_{3724}(53,\cdot)\) \(\chi_{3724}(261,\cdot)\) \(\chi_{3724}(333,\cdot)\) \(\chi_{3724}(345,\cdot)\) \(\chi_{3724}(401,\cdot)\) \(\chi_{3724}(585,\cdot)\) \(\chi_{3724}(697,\cdot)\) \(\chi_{3724}(793,\cdot)\) \(\chi_{3724}(865,\cdot)\) \(\chi_{3724}(877,\cdot)\) \(\chi_{3724}(933,\cdot)\) \(\chi_{3724}(1117,\cdot)\) \(\chi_{3724}(1229,\cdot)\) \(\chi_{3724}(1325,\cdot)\) \(\chi_{3724}(1397,\cdot)\) \(\chi_{3724}(1409,\cdot)\) \(\chi_{3724}(1465,\cdot)\) \(\chi_{3724}(1649,\cdot)\) \(\chi_{3724}(1761,\cdot)\) \(\chi_{3724}(1857,\cdot)\) \(\chi_{3724}(1997,\cdot)\) \(\chi_{3724}(2181,\cdot)\) \(\chi_{3724}(2293,\cdot)\) \(\chi_{3724}(2389,\cdot)\) \(\chi_{3724}(2461,\cdot)\) \(\chi_{3724}(2473,\cdot)\) \(\chi_{3724}(2825,\cdot)\) \(\chi_{3724}(2993,\cdot)\) \(\chi_{3724}(3005,\cdot)\) \(\chi_{3724}(3061,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((1863,3041,3137)\) → \((1,e\left(\frac{2}{21}\right),e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 3724 }(2825, a) \) \(-1\)\(1\)\(e\left(\frac{89}{126}\right)\)\(e\left(\frac{13}{63}\right)\)\(e\left(\frac{26}{63}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{67}{126}\right)\)\(e\left(\frac{115}{126}\right)\)\(e\left(\frac{10}{63}\right)\)\(e\left(\frac{11}{63}\right)\)\(e\left(\frac{26}{63}\right)\)\(e\left(\frac{5}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3724 }(2825,a) \;\) at \(\;a = \) e.g. 2