Properties

Label 2850.2197
Modulus $2850$
Conductor $475$
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(2850, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,51,50]))
 
pari: [g,chi] = znchar(Mod(2197,2850))
 

Basic properties

Modulus: \(2850\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(297,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2850.cg

\(\chi_{2850}(103,\cdot)\) \(\chi_{2850}(217,\cdot)\) \(\chi_{2850}(373,\cdot)\) \(\chi_{2850}(487,\cdot)\) \(\chi_{2850}(673,\cdot)\) \(\chi_{2850}(787,\cdot)\) \(\chi_{2850}(1513,\cdot)\) \(\chi_{2850}(1627,\cdot)\) \(\chi_{2850}(1813,\cdot)\) \(\chi_{2850}(1927,\cdot)\) \(\chi_{2850}(2083,\cdot)\) \(\chi_{2850}(2197,\cdot)\) \(\chi_{2850}(2383,\cdot)\) \(\chi_{2850}(2497,\cdot)\) \(\chi_{2850}(2653,\cdot)\) \(\chi_{2850}(2767,\cdot)\)

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

\((1901,1027,1351)\) → \((1,e\left(\frac{17}{20}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2850 }(2197, a) \) \(1\)\(1\)\(i\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2850 }(2197,a) \;\) at \(\;a = \) e.g. 2