Properties

Label 2550.11
Modulus $2550$
Conductor $1275$
Order $80$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,64,35]))
 
pari: [g,chi] = znchar(Mod(11,2550))
 

Basic properties

Modulus: \(2550\)
Conductor: \(1275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1275}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2550.cr

\(\chi_{2550}(11,\cdot)\) \(\chi_{2550}(41,\cdot)\) \(\chi_{2550}(71,\cdot)\) \(\chi_{2550}(131,\cdot)\) \(\chi_{2550}(311,\cdot)\) \(\chi_{2550}(371,\cdot)\) \(\chi_{2550}(431,\cdot)\) \(\chi_{2550}(521,\cdot)\) \(\chi_{2550}(581,\cdot)\) \(\chi_{2550}(641,\cdot)\) \(\chi_{2550}(821,\cdot)\) \(\chi_{2550}(881,\cdot)\) \(\chi_{2550}(911,\cdot)\) \(\chi_{2550}(941,\cdot)\) \(\chi_{2550}(1031,\cdot)\) \(\chi_{2550}(1061,\cdot)\) \(\chi_{2550}(1091,\cdot)\) \(\chi_{2550}(1331,\cdot)\) \(\chi_{2550}(1391,\cdot)\) \(\chi_{2550}(1421,\cdot)\) \(\chi_{2550}(1541,\cdot)\) \(\chi_{2550}(1571,\cdot)\) \(\chi_{2550}(1661,\cdot)\) \(\chi_{2550}(1841,\cdot)\) \(\chi_{2550}(1931,\cdot)\) \(\chi_{2550}(1961,\cdot)\) \(\chi_{2550}(2081,\cdot)\) \(\chi_{2550}(2111,\cdot)\) \(\chi_{2550}(2171,\cdot)\) \(\chi_{2550}(2411,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((851,1327,751)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{7}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2550 }(11, a) \) \(1\)\(1\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{29}{80}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{21}{40}\right)\)\(e\left(\frac{69}{80}\right)\)\(e\left(\frac{63}{80}\right)\)\(e\left(\frac{27}{80}\right)\)\(e\left(\frac{51}{80}\right)\)\(e\left(\frac{41}{80}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2550 }(11,a) \;\) at \(\;a = \) e.g. 2