Properties

Label 1380.857
Modulus $1380$
Conductor $345$
Order $44$
Real no
Primitive no
Minimal yes
Parity even

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([0,22,11,36]))
 
pari: [g,chi] = znchar(Mod(857,1380))
 

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 \(\chi_{345}(167,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1380.bu

\(\chi_{1380}(77,\cdot)\) \(\chi_{1380}(173,\cdot)\) \(\chi_{1380}(197,\cdot)\) \(\chi_{1380}(233,\cdot)\) \(\chi_{1380}(257,\cdot)\) \(\chi_{1380}(317,\cdot)\) \(\chi_{1380}(353,\cdot)\) \(\chi_{1380}(377,\cdot)\) \(\chi_{1380}(473,\cdot)\) \(\chi_{1380}(533,\cdot)\) \(\chi_{1380}(593,\cdot)\) \(\chi_{1380}(653,\cdot)\) \(\chi_{1380}(857,\cdot)\) \(\chi_{1380}(1037,\cdot)\) \(\chi_{1380}(1097,\cdot)\) \(\chi_{1380}(1133,\cdot)\) \(\chi_{1380}(1277,\cdot)\) \(\chi_{1380}(1313,\cdot)\) \(\chi_{1380}(1337,\cdot)\) \(\chi_{1380}(1373,\cdot)\)

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: 44.44.10759477646680772288556262812807675479811013525016604751482416550745256245136260986328125.1

Values on generators

\((691,461,277,1201)\) → \((1,-1,i,e\left(\frac{9}{11}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{37}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1380 }(857,a) \;\) at \(\;a = \) e.g. 2