Properties

Label 1380.53
Modulus $1380$
Conductor $345$
Order $44$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1380, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,22,33,38]))
 
pari: [g,chi] = znchar(Mod(53,1380))
 

Basic properties

Modulus: \(1380\)
Conductor: \(345\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{345}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1380.bo

\(\chi_{1380}(17,\cdot)\) \(\chi_{1380}(53,\cdot)\) \(\chi_{1380}(113,\cdot)\) \(\chi_{1380}(293,\cdot)\) \(\chi_{1380}(497,\cdot)\) \(\chi_{1380}(557,\cdot)\) \(\chi_{1380}(617,\cdot)\) \(\chi_{1380}(677,\cdot)\) \(\chi_{1380}(773,\cdot)\) \(\chi_{1380}(797,\cdot)\) \(\chi_{1380}(833,\cdot)\) \(\chi_{1380}(893,\cdot)\) \(\chi_{1380}(917,\cdot)\) \(\chi_{1380}(953,\cdot)\) \(\chi_{1380}(977,\cdot)\) \(\chi_{1380}(1073,\cdot)\) \(\chi_{1380}(1157,\cdot)\) \(\chi_{1380}(1193,\cdot)\) \(\chi_{1380}(1217,\cdot)\) \(\chi_{1380}(1253,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Fixed field: 44.0.5691763675094128540646263027975260328820026154733783913534198355344240553677082061767578125.1

Values on generators

\((691,461,277,1201)\) → \((1,-1,-i,e\left(\frac{19}{22}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(-1\)\(1\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{25}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1380 }(53,a) \;\) at \(\;a = \) e.g. 2