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Properties

Label 2321.1813
Modulus $2321$
Conductor $2321$
Order $35$
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(2321, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([42,62]))
 
Copy content gp:[g,chi] = znchar(Mod(1813, 2321))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2321.1813");
 

Basic properties

Modulus: \(2321\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2321\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(35\)
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 2321.cm

\(\chi_{2321}(64,\cdot)\) \(\chi_{2321}(236,\cdot)\) \(\chi_{2321}(290,\cdot)\) \(\chi_{2321}(324,\cdot)\) \(\chi_{2321}(394,\cdot)\) \(\chi_{2321}(433,\cdot)\) \(\chi_{2321}(531,\cdot)\) \(\chi_{2321}(544,\cdot)\) \(\chi_{2321}(729,\cdot)\) \(\chi_{2321}(1028,\cdot)\) \(\chi_{2321}(1060,\cdot)\) \(\chi_{2321}(1120,\cdot)\) \(\chi_{2321}(1131,\cdot)\) \(\chi_{2321}(1169,\cdot)\) \(\chi_{2321}(1224,\cdot)\) \(\chi_{2321}(1490,\cdot)\) \(\chi_{2321}(1775,\cdot)\) \(\chi_{2321}(1809,\cdot)\) \(\chi_{2321}(1813,\cdot)\) \(\chi_{2321}(2050,\cdot)\) \(\chi_{2321}(2192,\cdot)\) \(\chi_{2321}(2253,\cdot)\) \(\chi_{2321}(2303,\cdot)\) \(\chi_{2321}(2313,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 35 polynomial

Values on generators

\((1267,1057)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{31}{35}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 2321 }(1813, a) \) \(1\)\(1\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{27}{35}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{6}{7}\right)\)
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)
 
\( \chi_{ 2321 }(1813,a) \;\) at \(\;a = \) e.g. 2