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Properties

Label 31753.26616
Modulus $31753$
Conductor $31753$
Order $56$
Real no
Primitive yes
Minimal yes
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(31753, base_ring=CyclotomicField(56)) M = H._module chi = DirichletCharacter(H, M([51,20]))
 
Copy content gp:[g,chi] = znchar(Mod(26616, 31753))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31753.26616");
 

Basic properties

Modulus: \(31753\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(31753\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(56\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 31753.hi

\(\chi_{31753}(1908,\cdot)\) \(\chi_{31753}(3123,\cdot)\) \(\chi_{31753}(4809,\cdot)\) \(\chi_{31753}(5280,\cdot)\) \(\chi_{31753}(6579,\cdot)\) \(\chi_{31753}(7422,\cdot)\) \(\chi_{31753}(8351,\cdot)\) \(\chi_{31753}(14814,\cdot)\) \(\chi_{31753}(14834,\cdot)\) \(\chi_{31753}(15346,\cdot)\) \(\chi_{31753}(17803,\cdot)\) \(\chi_{31753}(18206,\cdot)\) \(\chi_{31753}(20123,\cdot)\) \(\chi_{31753}(23495,\cdot)\) \(\chi_{31753}(23704,\cdot)\) \(\chi_{31753}(24547,\cdot)\) \(\chi_{31753}(25687,\cdot)\) \(\chi_{31753}(26530,\cdot)\) \(\chi_{31753}(26616,\cdot)\) \(\chi_{31753}(28272,\cdot)\) \(\chi_{31753}(28302,\cdot)\) \(\chi_{31753}(28413,\cdot)\) \(\chi_{31753}(30099,\cdot)\) \(\chi_{31753}(30448,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((20795,10962)\) → \((e\left(\frac{51}{56}\right),e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 31753 }(26616, a) \) \(1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{15}{56}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{56}\right)\)\(e\left(\frac{3}{56}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{45}{56}\right)\)\(-i\)
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)
 
\( \chi_{ 31753 }(26616,a) \;\) at \(\;a = \) e.g. 2