Properties

Label 5292.193
Modulus $5292$
Conductor $1323$
Order $63$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,14,66]))
 
pari: [g,chi] = znchar(Mod(193,5292))
 

Basic properties

Modulus: \(5292\)
Conductor: \(1323\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(63\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1323}(193,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5292.eb

\(\chi_{5292}(193,\cdot)\) \(\chi_{5292}(205,\cdot)\) \(\chi_{5292}(445,\cdot)\) \(\chi_{5292}(457,\cdot)\) \(\chi_{5292}(697,\cdot)\) \(\chi_{5292}(709,\cdot)\) \(\chi_{5292}(1201,\cdot)\) \(\chi_{5292}(1213,\cdot)\) \(\chi_{5292}(1453,\cdot)\) \(\chi_{5292}(1465,\cdot)\) \(\chi_{5292}(1705,\cdot)\) \(\chi_{5292}(1717,\cdot)\) \(\chi_{5292}(1957,\cdot)\) \(\chi_{5292}(1969,\cdot)\) \(\chi_{5292}(2209,\cdot)\) \(\chi_{5292}(2221,\cdot)\) \(\chi_{5292}(2461,\cdot)\) \(\chi_{5292}(2473,\cdot)\) \(\chi_{5292}(2965,\cdot)\) \(\chi_{5292}(2977,\cdot)\) \(\chi_{5292}(3217,\cdot)\) \(\chi_{5292}(3229,\cdot)\) \(\chi_{5292}(3469,\cdot)\) \(\chi_{5292}(3481,\cdot)\) \(\chi_{5292}(3721,\cdot)\) \(\chi_{5292}(3733,\cdot)\) \(\chi_{5292}(3973,\cdot)\) \(\chi_{5292}(3985,\cdot)\) \(\chi_{5292}(4225,\cdot)\) \(\chi_{5292}(4237,\cdot)\) ...

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 63 polynomial

Values on generators

\((2647,785,1081)\) → \((1,e\left(\frac{1}{9}\right),e\left(\frac{11}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 5292 }(193, a) \) \(1\)\(1\)\(e\left(\frac{47}{63}\right)\)\(e\left(\frac{25}{63}\right)\)\(e\left(\frac{11}{63}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{63}\right)\)\(e\left(\frac{31}{63}\right)\)\(e\left(\frac{34}{63}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5292 }(193,a) \;\) at \(\;a = \) e.g. 2