Properties

Label 1254.787
Modulus $1254$
Conductor $209$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1254, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,27,5]))
 
pari: [g,chi] = znchar(Mod(787,1254))
 

Basic properties

Modulus: \(1254\)
Conductor: \(209\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{209}(160,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1254.bn

\(\chi_{1254}(145,\cdot)\) \(\chi_{1254}(217,\cdot)\) \(\chi_{1254}(259,\cdot)\) \(\chi_{1254}(601,\cdot)\) \(\chi_{1254}(673,\cdot)\) \(\chi_{1254}(787,\cdot)\) \(\chi_{1254}(943,\cdot)\) \(\chi_{1254}(1129,\cdot)\)

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: 30.30.1220232317838205647399552173000517992590495082743137882605129.1

Values on generators

\((419,343,1123)\) → \((1,e\left(\frac{9}{10}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 1254 }(787, a) \) \(1\)\(1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1254 }(787,a) \;\) at \(\;a = \) e.g. 2