Properties

Label 2760.203
Modulus $2760$
Conductor $2760$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,22,22,33,30]))
 
pari: [g,chi] = znchar(Mod(203,2760))
 

Basic properties

Modulus: \(2760\)
Conductor: \(2760\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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 2760.df

\(\chi_{2760}(83,\cdot)\) \(\chi_{2760}(107,\cdot)\) \(\chi_{2760}(203,\cdot)\) \(\chi_{2760}(227,\cdot)\) \(\chi_{2760}(467,\cdot)\) \(\chi_{2760}(563,\cdot)\) \(\chi_{2760}(707,\cdot)\) \(\chi_{2760}(803,\cdot)\) \(\chi_{2760}(1187,\cdot)\) \(\chi_{2760}(1307,\cdot)\) \(\chi_{2760}(1523,\cdot)\) \(\chi_{2760}(1643,\cdot)\) \(\chi_{2760}(1667,\cdot)\) \(\chi_{2760}(1763,\cdot)\) \(\chi_{2760}(1883,\cdot)\) \(\chi_{2760}(1907,\cdot)\) \(\chi_{2760}(2123,\cdot)\) \(\chi_{2760}(2363,\cdot)\) \(\chi_{2760}(2627,\cdot)\) \(\chi_{2760}(2747,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((2071,1381,1841,1657,1201)\) → \((-1,-1,-1,-i,e\left(\frac{15}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2760 }(203, a) \) \(1\)\(1\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{29}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2760 }(203,a) \;\) at \(\;a = \) e.g. 2