Properties

Label 4030.2297
Modulus $4030$
Conductor $2015$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4030, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,40,2]))
 
pari: [g,chi] = znchar(Mod(2297,4030))
 

Basic properties

Modulus: \(4030\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2015}(282,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4030.gy

\(\chi_{4030}(393,\cdot)\) \(\chi_{4030}(477,\cdot)\) \(\chi_{4030}(737,\cdot)\) \(\chi_{4030}(757,\cdot)\) \(\chi_{4030}(1257,\cdot)\) \(\chi_{4030}(1283,\cdot)\) \(\chi_{4030}(1543,\cdot)\) \(\chi_{4030}(1563,\cdot)\) \(\chi_{4030}(2057,\cdot)\) \(\chi_{4030}(2063,\cdot)\) \(\chi_{4030}(2297,\cdot)\) \(\chi_{4030}(2863,\cdot)\) \(\chi_{4030}(2967,\cdot)\) \(\chi_{4030}(3103,\cdot)\) \(\chi_{4030}(3617,\cdot)\) \(\chi_{4030}(3773,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((807,1861,2731)\) → \((i,e\left(\frac{2}{3}\right),e\left(\frac{1}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4030 }(2297, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4030 }(2297,a) \;\) at \(\;a = \) e.g. 2