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Properties

Label 312325.3249
Modulus $312325$
Conductor $62465$
Order $186$
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(312325, base_ring=CyclotomicField(186)) M = H._module chi = DirichletCharacter(H, M([93,93,86]))
 
Copy content gp:[g,chi] = znchar(Mod(3249, 312325))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("312325.3249");
 

Basic properties

Modulus: \(312325\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(62465\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(186\)
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 \(\chi_{62465}(3249,\cdot)\)
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 312325.bcc

\(\chi_{312325}(3249,\cdot)\) \(\chi_{312325}(6174,\cdot)\) \(\chi_{312325}(13324,\cdot)\) \(\chi_{312325}(16249,\cdot)\) \(\chi_{312325}(23399,\cdot)\) \(\chi_{312325}(26324,\cdot)\) \(\chi_{312325}(33474,\cdot)\) \(\chi_{312325}(36399,\cdot)\) \(\chi_{312325}(43549,\cdot)\) \(\chi_{312325}(46474,\cdot)\) \(\chi_{312325}(53624,\cdot)\) \(\chi_{312325}(56549,\cdot)\) \(\chi_{312325}(63699,\cdot)\) \(\chi_{312325}(66624,\cdot)\) \(\chi_{312325}(73774,\cdot)\) \(\chi_{312325}(76699,\cdot)\) \(\chi_{312325}(83849,\cdot)\) \(\chi_{312325}(86774,\cdot)\) \(\chi_{312325}(93924,\cdot)\) \(\chi_{312325}(96849,\cdot)\) \(\chi_{312325}(103999,\cdot)\) \(\chi_{312325}(106924,\cdot)\) \(\chi_{312325}(114074,\cdot)\) \(\chi_{312325}(116999,\cdot)\) \(\chi_{312325}(124149,\cdot)\) \(\chi_{312325}(127074,\cdot)\) \(\chi_{312325}(134224,\cdot)\) \(\chi_{312325}(137149,\cdot)\) \(\chi_{312325}(144299,\cdot)\) \(\chi_{312325}(147224,\cdot)\) ...

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: $\Q(\zeta_{93})$
Fixed field: Number field defined by a degree 186 polynomial (not computed)

Values on generators

\((87452,24026,89376)\) → \((-1,-1,e\left(\frac{43}{93}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 312325 }(3249, a) \) \(1\)\(1\)\(e\left(\frac{16}{31}\right)\)\(e\left(\frac{179}{186}\right)\)\(e\left(\frac{1}{31}\right)\)\(e\left(\frac{89}{186}\right)\)\(e\left(\frac{22}{93}\right)\)\(e\left(\frac{17}{31}\right)\)\(e\left(\frac{86}{93}\right)\)\(e\left(\frac{157}{186}\right)\)\(e\left(\frac{185}{186}\right)\)\(e\left(\frac{70}{93}\right)\)
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)
 
\( \chi_{ 312325 }(3249,a) \;\) at \(\;a = \) e.g. 2