↑

Properties

Label 1385.99
Modulus $1385$
Conductor $1385$
Order $276$
Real no
Primitive yes
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(1385, base_ring=CyclotomicField(276)) M = H._module chi = DirichletCharacter(H, M([138,107]))
 
Copy content gp:[g,chi] = znchar(Mod(99, 1385))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1385.99");
 

Basic properties

Modulus: \(1385\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1385\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(276\)
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: yes
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 1385.bj

\(\chi_{1385}(14,\cdot)\) \(\chi_{1385}(24,\cdot)\) \(\chi_{1385}(44,\cdot)\) \(\chi_{1385}(94,\cdot)\) \(\chi_{1385}(99,\cdot)\) \(\chi_{1385}(114,\cdot)\) \(\chi_{1385}(119,\cdot)\) \(\chi_{1385}(124,\cdot)\) \(\chi_{1385}(134,\cdot)\) \(\chi_{1385}(174,\cdot)\) \(\chi_{1385}(179,\cdot)\) \(\chi_{1385}(184,\cdot)\) \(\chi_{1385}(199,\cdot)\) \(\chi_{1385}(209,\cdot)\) \(\chi_{1385}(219,\cdot)\) \(\chi_{1385}(224,\cdot)\) \(\chi_{1385}(234,\cdot)\) \(\chi_{1385}(259,\cdot)\) \(\chi_{1385}(294,\cdot)\) \(\chi_{1385}(349,\cdot)\) \(\chi_{1385}(354,\cdot)\) \(\chi_{1385}(374,\cdot)\) \(\chi_{1385}(384,\cdot)\) \(\chi_{1385}(404,\cdot)\) \(\chi_{1385}(414,\cdot)\) \(\chi_{1385}(419,\cdot)\) \(\chi_{1385}(439,\cdot)\) \(\chi_{1385}(444,\cdot)\) \(\chi_{1385}(449,\cdot)\) \(\chi_{1385}(474,\cdot)\) ...

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: $\Q(\zeta_{276})$
Fixed field: Number field defined by a degree 276 polynomial (not computed)

Values on generators

\((832,836)\) → \((-1,e\left(\frac{107}{276}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1385 }(99, a) \) \(-1\)\(1\)\(e\left(\frac{45}{92}\right)\)\(e\left(\frac{53}{138}\right)\)\(e\left(\frac{45}{46}\right)\)\(e\left(\frac{241}{276}\right)\)\(e\left(\frac{2}{69}\right)\)\(e\left(\frac{43}{92}\right)\)\(e\left(\frac{53}{69}\right)\)\(e\left(\frac{197}{276}\right)\)\(e\left(\frac{25}{69}\right)\)\(e\left(\frac{13}{23}\right)\)
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)
 
\( \chi_{ 1385 }(99,a) \;\) at \(\;a = \) e.g. 2