Properties

Label 1400.227
Modulus $1400$
Conductor $1400$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,30,3,10]))
 
pari: [g,chi] = znchar(Mod(227,1400))
 

Basic properties

Modulus: \(1400\)
Conductor: \(1400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1400.dq

\(\chi_{1400}(3,\cdot)\) \(\chi_{1400}(187,\cdot)\) \(\chi_{1400}(227,\cdot)\) \(\chi_{1400}(283,\cdot)\) \(\chi_{1400}(467,\cdot)\) \(\chi_{1400}(523,\cdot)\) \(\chi_{1400}(563,\cdot)\) \(\chi_{1400}(747,\cdot)\) \(\chi_{1400}(787,\cdot)\) \(\chi_{1400}(803,\cdot)\) \(\chi_{1400}(1027,\cdot)\) \(\chi_{1400}(1067,\cdot)\) \(\chi_{1400}(1083,\cdot)\) \(\chi_{1400}(1123,\cdot)\) \(\chi_{1400}(1347,\cdot)\) \(\chi_{1400}(1363,\cdot)\)

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

\((351,701,1177,801)\) → \((-1,-1,e\left(\frac{1}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 1400 }(227, a) \) \(-1\)\(1\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1400 }(227,a) \;\) at \(\;a = \) e.g. 2