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Properties

Label 309680.61937
Modulus $309680$
Conductor $5$
Order $4$
Real no
Primitive no
Minimal no
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(309680, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,0,1,0,0]))
 
Copy content pari:[g,chi] = znchar(Mod(61937,309680))
 

Basic properties

Modulus: \(309680\)
Conductor: \(5\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{5}(2,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 309680.cr

\(\chi_{309680}(61937,\cdot)\) \(\chi_{309680}(123873,\cdot)\)

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

Related number fields

Field of values: \(\mathbb{Q}(i)\)
Fixed field: \(\Q(\zeta_{5})\)

Values on generators

\((193551,232261,61937,297041,82321)\) → \((1,1,i,1,1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 309680 }(61937, a) \) \(-1\)\(1\)\(-i\)\(-1\)\(1\)\(-i\)\(i\)\(-1\)\(-i\)\(i\)\(-1\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 309680 }(61937,a) \;\) at \(\;a = \) e.g. 2