↑

Properties

Label 312325.15182
Modulus $312325$
Conductor $62465$
Order $1860$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

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(1860)) M = H._module chi = DirichletCharacter(H, M([465,1085,54]))
 
Copy content gp:[g,chi] = znchar(Mod(15182, 312325))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("312325.15182");
 

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: \(1860\)
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 \(\chi_{62465}(15182,\cdot)\)
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: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 312325.cak

\(\chi_{312325}(232,\cdot)\) \(\chi_{312325}(418,\cdot)\) \(\chi_{312325}(643,\cdot)\) \(\chi_{312325}(1207,\cdot)\) \(\chi_{312325}(1393,\cdot)\) \(\chi_{312325}(2507,\cdot)\) \(\chi_{312325}(2693,\cdot)\) \(\chi_{312325}(4882,\cdot)\) \(\chi_{312325}(5107,\cdot)\) \(\chi_{312325}(5293,\cdot)\) \(\chi_{312325}(5843,\cdot)\) \(\chi_{312325}(5857,\cdot)\) \(\chi_{312325}(6818,\cdot)\) \(\chi_{312325}(9757,\cdot)\) \(\chi_{312325}(10307,\cdot)\) \(\chi_{312325}(10493,\cdot)\) \(\chi_{312325}(10718,\cdot)\) \(\chi_{312325}(11282,\cdot)\) \(\chi_{312325}(11468,\cdot)\) \(\chi_{312325}(12582,\cdot)\) \(\chi_{312325}(12768,\cdot)\) \(\chi_{312325}(14957,\cdot)\) \(\chi_{312325}(15182,\cdot)\) \(\chi_{312325}(15368,\cdot)\) \(\chi_{312325}(15918,\cdot)\) \(\chi_{312325}(15932,\cdot)\) \(\chi_{312325}(16893,\cdot)\) \(\chi_{312325}(17232,\cdot)\) \(\chi_{312325}(18193,\cdot)\) \(\chi_{312325}(19832,\cdot)\) ...

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: $\Q(\zeta_{1860})$
Fixed field: Number field defined by a degree 1860 polynomial (not computed)

Values on generators

\((87452,24026,89376)\) → \((i,e\left(\frac{7}{12}\right),e\left(\frac{9}{310}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 312325 }(15182, a) \) \(-1\)\(1\)\(e\left(\frac{883}{930}\right)\)\(e\left(\frac{209}{1860}\right)\)\(e\left(\frac{418}{465}\right)\)\(e\left(\frac{23}{372}\right)\)\(e\left(\frac{358}{465}\right)\)\(e\left(\frac{263}{310}\right)\)\(e\left(\frac{209}{930}\right)\)\(e\left(\frac{857}{1860}\right)\)\(e\left(\frac{7}{620}\right)\)\(e\left(\frac{223}{310}\right)\)
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)
 
\( \chi_{ 312325 }(15182,a) \;\) at \(\;a = \) e.g. 2