Properties

Label 1476.305
Modulus $1476$
Conductor $123$
Order $10$
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(1476, base_ring=CyclotomicField(10))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,5,4]))
 
pari: [g,chi] = znchar(Mod(305,1476))
 

Basic properties

Modulus: \(1476\)
Conductor: \(123\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{123}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1476.bc

\(\chi_{1476}(305,\cdot)\) \(\chi_{1476}(953,\cdot)\) \(\chi_{1476}(1205,\cdot)\) \(\chi_{1476}(1349,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.0.1940336830676403.1

Values on generators

\((739,821,1441)\) → \((1,-1,e\left(\frac{2}{5}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1476 }(305, a) \) \(-1\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1476 }(305,a) \;\) at \(\;a = \) e.g. 2