Properties

Label 4030.693
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([45,10,46]))
 
pari: [g,chi] = znchar(Mod(693,4030))
 

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 \(\chi_{2015}(693,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4030.gm

\(\chi_{4030}(127,\cdot)\) \(\chi_{4030}(693,\cdot)\) \(\chi_{4030}(797,\cdot)\) \(\chi_{4030}(933,\cdot)\) \(\chi_{4030}(1447,\cdot)\) \(\chi_{4030}(1603,\cdot)\) \(\chi_{4030}(2253,\cdot)\) \(\chi_{4030}(2337,\cdot)\) \(\chi_{4030}(2597,\cdot)\) \(\chi_{4030}(2617,\cdot)\) \(\chi_{4030}(3117,\cdot)\) \(\chi_{4030}(3143,\cdot)\) \(\chi_{4030}(3403,\cdot)\) \(\chi_{4030}(3423,\cdot)\) \(\chi_{4030}(3917,\cdot)\) \(\chi_{4030}(3923,\cdot)\)

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

\((807,1861,2731)\) → \((-i,e\left(\frac{1}{6}\right),e\left(\frac{23}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4030 }(693, a) \) \(1\)\(1\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4030 }(693,a) \;\) at \(\;a = \) e.g. 2