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Properties

Label 8464.7805
Modulus $8464$
Conductor $368$
Order $44$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(44)) M = H._module chi = DirichletCharacter(H, M([0,33,12]))
 
Copy content gp:[g,chi] = znchar(Mod(7805, 8464))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8464.7805");
 

Basic properties

Modulus: \(8464\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(368\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(44\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from \(\chi_{368}(77,\cdot)\)
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: no
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 8464.x

\(\chi_{8464}(501,\cdot)\) \(\chi_{8464}(1757,\cdot)\) \(\chi_{8464}(1853,\cdot)\) \(\chi_{8464}(2053,\cdot)\) \(\chi_{8464}(2293,\cdot)\) \(\chi_{8464}(3429,\cdot)\) \(\chi_{8464}(3573,\cdot)\) \(\chi_{8464}(3661,\cdot)\) \(\chi_{8464}(3821,\cdot)\) \(\chi_{8464}(4037,\cdot)\) \(\chi_{8464}(4733,\cdot)\) \(\chi_{8464}(5989,\cdot)\) \(\chi_{8464}(6085,\cdot)\) \(\chi_{8464}(6285,\cdot)\) \(\chi_{8464}(6525,\cdot)\) \(\chi_{8464}(7661,\cdot)\) \(\chi_{8464}(7805,\cdot)\) \(\chi_{8464}(7893,\cdot)\) \(\chi_{8464}(8053,\cdot)\) \(\chi_{8464}(8269,\cdot)\)

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Fixed field: 44.44.7829660228065619245582194641412012312544945884150589900838471630076269829766255604192509952.1

Values on generators

\((7407,2117,6353)\) → \((1,-i,e\left(\frac{3}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8464 }(7805, a) \) \(1\)\(1\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{13}{44}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 8464 }(7805,a) \;\) at \(\;a = \) e.g. 2