Properties

Label 1337.544
Modulus $1337$
Conductor $1337$
Order $570$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1337, base_ring=CyclotomicField(570))
 
M = H._module
 
chi = DirichletCharacter(H, M([475,384]))
 
pari: [g,chi] = znchar(Mod(544,1337))
 

Basic properties

Modulus: \(1337\)
Conductor: \(1337\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(570\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1337.bf

\(\chi_{1337}(3,\cdot)\) \(\chi_{1337}(10,\cdot)\) \(\chi_{1337}(12,\cdot)\) \(\chi_{1337}(17,\cdot)\) \(\chi_{1337}(24,\cdot)\) \(\chi_{1337}(26,\cdot)\) \(\chi_{1337}(40,\cdot)\) \(\chi_{1337}(45,\cdot)\) \(\chi_{1337}(54,\cdot)\) \(\chi_{1337}(59,\cdot)\) \(\chi_{1337}(68,\cdot)\) \(\chi_{1337}(75,\cdot)\) \(\chi_{1337}(80,\cdot)\) \(\chi_{1337}(96,\cdot)\) \(\chi_{1337}(103,\cdot)\) \(\chi_{1337}(108,\cdot)\) \(\chi_{1337}(115,\cdot)\) \(\chi_{1337}(117,\cdot)\) \(\chi_{1337}(129,\cdot)\) \(\chi_{1337}(138,\cdot)\) \(\chi_{1337}(194,\cdot)\) \(\chi_{1337}(199,\cdot)\) \(\chi_{1337}(201,\cdot)\) \(\chi_{1337}(206,\cdot)\) \(\chi_{1337}(208,\cdot)\) \(\chi_{1337}(215,\cdot)\) \(\chi_{1337}(234,\cdot)\) \(\chi_{1337}(236,\cdot)\) \(\chi_{1337}(241,\cdot)\) \(\chi_{1337}(250,\cdot)\) ...

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: $\Q(\zeta_{285})$
Fixed field: Number field defined by a degree 570 polynomial (not computed)

Values on generators

\((192,974)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{64}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1337 }(544, a) \) \(-1\)\(1\)\(e\left(\frac{88}{285}\right)\)\(e\left(\frac{559}{570}\right)\)\(e\left(\frac{176}{285}\right)\)\(e\left(\frac{97}{114}\right)\)\(e\left(\frac{11}{38}\right)\)\(e\left(\frac{88}{95}\right)\)\(e\left(\frac{274}{285}\right)\)\(e\left(\frac{91}{570}\right)\)\(e\left(\frac{34}{57}\right)\)\(e\left(\frac{341}{570}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1337 }(544,a) \;\) at \(\;a = \) e.g. 2