Properties

Label 4030.11
Modulus $4030$
Conductor $403$
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([0,35,46]))
 
pari: [g,chi] = znchar(Mod(11,4030))
 

Basic properties

Modulus: \(4030\)
Conductor: \(403\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{403}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4030.hq

\(\chi_{4030}(11,\cdot)\) \(\chi_{4030}(241,\cdot)\) \(\chi_{4030}(301,\cdot)\) \(\chi_{4030}(761,\cdot)\) \(\chi_{4030}(921,\cdot)\) \(\chi_{4030}(951,\cdot)\) \(\chi_{4030}(1571,\cdot)\) \(\chi_{4030}(1801,\cdot)\) \(\chi_{4030}(2121,\cdot)\) \(\chi_{4030}(2151,\cdot)\) \(\chi_{4030}(2411,\cdot)\) \(\chi_{4030}(2741,\cdot)\) \(\chi_{4030}(2931,\cdot)\) \(\chi_{4030}(3421,\cdot)\) \(\chi_{4030}(3971,\cdot)\) \(\chi_{4030}(4011,\cdot)\)

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

\((807,1861,2731)\) → \((1,e\left(\frac{7}{12}\right),e\left(\frac{23}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4030 }(11, a) \) \(1\)\(1\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4030 }(11,a) \;\) at \(\;a = \) e.g. 2