Properties

Label 1380.157
Modulus $1380$
Conductor $115$
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([0,0,11,30]))
 
pari: [g,chi] = znchar(Mod(157,1380))
 

Basic properties

Modulus: \(1380\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(42,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1380.bs

\(\chi_{1380}(37,\cdot)\) \(\chi_{1380}(97,\cdot)\) \(\chi_{1380}(157,\cdot)\) \(\chi_{1380}(217,\cdot)\) \(\chi_{1380}(313,\cdot)\) \(\chi_{1380}(337,\cdot)\) \(\chi_{1380}(373,\cdot)\) \(\chi_{1380}(433,\cdot)\) \(\chi_{1380}(457,\cdot)\) \(\chi_{1380}(493,\cdot)\) \(\chi_{1380}(517,\cdot)\) \(\chi_{1380}(613,\cdot)\) \(\chi_{1380}(697,\cdot)\) \(\chi_{1380}(733,\cdot)\) \(\chi_{1380}(757,\cdot)\) \(\chi_{1380}(793,\cdot)\) \(\chi_{1380}(937,\cdot)\) \(\chi_{1380}(973,\cdot)\) \(\chi_{1380}(1033,\cdot)\) \(\chi_{1380}(1213,\cdot)\)

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: \(\Q(\zeta_{115})^+\)

Values on generators

\((691,461,277,1201)\) → \((1,1,i,e\left(\frac{15}{22}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{7}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1380 }(157,a) \;\) at \(\;a = \) e.g. 2