Properties

Label 4200.53
Modulus $4200$
Conductor $4200$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4200, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,30,30,21,40]))
 
pari: [g,chi] = znchar(Mod(53,4200))
 

Basic properties

Modulus: \(4200\)
Conductor: \(4200\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4200.hj

\(\chi_{4200}(53,\cdot)\) \(\chi_{4200}(317,\cdot)\) \(\chi_{4200}(653,\cdot)\) \(\chi_{4200}(1397,\cdot)\) \(\chi_{4200}(1733,\cdot)\) \(\chi_{4200}(1997,\cdot)\) \(\chi_{4200}(2237,\cdot)\) \(\chi_{4200}(2333,\cdot)\) \(\chi_{4200}(2573,\cdot)\) \(\chi_{4200}(2837,\cdot)\) \(\chi_{4200}(3077,\cdot)\) \(\chi_{4200}(3173,\cdot)\) \(\chi_{4200}(3413,\cdot)\) \(\chi_{4200}(3677,\cdot)\) \(\chi_{4200}(3917,\cdot)\) \(\chi_{4200}(4013,\cdot)\)

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

\((3151,2101,2801,1177,3601)\) → \((1,-1,-1,e\left(\frac{7}{20}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4200 }(53, a) \) \(1\)\(1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{9}{10}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4200 }(53,a) \;\) at \(\;a = \) e.g. 2