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Properties

Label 2611.1756
Modulus $2611$
Conductor $2611$
Order $62$
Real no
Primitive yes
Minimal yes
Parity odd

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(2611, base_ring=CyclotomicField(62)) M = H._module chi = DirichletCharacter(H, M([31,19]))
 
Copy content gp:[g,chi] = znchar(Mod(1756, 2611))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2611.1756");
 

Basic properties

Modulus: \(2611\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2611\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(62\)
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 2611.bh

\(\chi_{2611}(13,\cdot)\) \(\chi_{2611}(27,\cdot)\) \(\chi_{2611}(55,\cdot)\) \(\chi_{2611}(160,\cdot)\) \(\chi_{2611}(510,\cdot)\) \(\chi_{2611}(531,\cdot)\) \(\chi_{2611}(671,\cdot)\) \(\chi_{2611}(734,\cdot)\) \(\chi_{2611}(902,\cdot)\) \(\chi_{2611}(930,\cdot)\) \(\chi_{2611}(965,\cdot)\) \(\chi_{2611}(1000,\cdot)\) \(\chi_{2611}(1028,\cdot)\) \(\chi_{2611}(1070,\cdot)\) \(\chi_{2611}(1126,\cdot)\) \(\chi_{2611}(1203,\cdot)\) \(\chi_{2611}(1329,\cdot)\) \(\chi_{2611}(1406,\cdot)\) \(\chi_{2611}(1462,\cdot)\) \(\chi_{2611}(1644,\cdot)\) \(\chi_{2611}(1721,\cdot)\) \(\chi_{2611}(1756,\cdot)\) \(\chi_{2611}(1882,\cdot)\) \(\chi_{2611}(1896,\cdot)\) \(\chi_{2611}(1952,\cdot)\) \(\chi_{2611}(2127,\cdot)\) \(\chi_{2611}(2197,\cdot)\) \(\chi_{2611}(2260,\cdot)\) \(\chi_{2611}(2302,\cdot)\) \(\chi_{2611}(2442,\cdot)\)

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: $\Q(\zeta_{31})$
Fixed field: Number field defined by a degree 62 polynomial

Values on generators

\((374,1121)\) → \((-1,e\left(\frac{19}{62}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 2611 }(1756, a) \) \(-1\)\(1\)\(e\left(\frac{19}{62}\right)\)\(e\left(\frac{27}{62}\right)\)\(e\left(\frac{19}{31}\right)\)\(e\left(\frac{9}{31}\right)\)\(e\left(\frac{23}{31}\right)\)\(e\left(\frac{57}{62}\right)\)\(e\left(\frac{27}{31}\right)\)\(e\left(\frac{37}{62}\right)\)\(e\left(\frac{5}{62}\right)\)\(e\left(\frac{3}{62}\right)\)
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)
 
\( \chi_{ 2611 }(1756,a) \;\) at \(\;a = \) e.g. 2