Properties

Label 1575.1306
Modulus $1575$
Conductor $175$
Order $15$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,12,20]))
 
pari: [g,chi] = znchar(Mod(1306,1575))
 

Basic properties

Modulus: \(1575\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{175}(81,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1575.cs

\(\chi_{1575}(46,\cdot)\) \(\chi_{1575}(361,\cdot)\) \(\chi_{1575}(541,\cdot)\) \(\chi_{1575}(856,\cdot)\) \(\chi_{1575}(991,\cdot)\) \(\chi_{1575}(1171,\cdot)\) \(\chi_{1575}(1306,\cdot)\) \(\chi_{1575}(1486,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 15 polynomial

Values on generators

\((1226,127,451)\) → \((1,e\left(\frac{2}{5}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 1575 }(1306, a) \) \(1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1575 }(1306,a) \;\) at \(\;a = \) e.g. 2