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Properties

Label 2844.1219
Modulus $2844$
Conductor $2844$
Order $78$
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(2844, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([39,26,25]))
 
Copy content gp:[g,chi] = znchar(Mod(1219, 2844))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2844.1219");
 

Basic properties

Modulus: \(2844\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2844\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(78\)
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 2844.cc

\(\chi_{2844}(7,\cdot)\) \(\chi_{2844}(187,\cdot)\) \(\chi_{2844}(319,\cdot)\) \(\chi_{2844}(607,\cdot)\) \(\chi_{2844}(943,\cdot)\) \(\chi_{2844}(1087,\cdot)\) \(\chi_{2844}(1159,\cdot)\) \(\chi_{2844}(1183,\cdot)\) \(\chi_{2844}(1219,\cdot)\) \(\chi_{2844}(1303,\cdot)\) \(\chi_{2844}(1327,\cdot)\) \(\chi_{2844}(1339,\cdot)\) \(\chi_{2844}(1411,\cdot)\) \(\chi_{2844}(1615,\cdot)\) \(\chi_{2844}(1627,\cdot)\) \(\chi_{2844}(1687,\cdot)\) \(\chi_{2844}(1939,\cdot)\) \(\chi_{2844}(2023,\cdot)\) \(\chi_{2844}(2203,\cdot)\) \(\chi_{2844}(2407,\cdot)\) \(\chi_{2844}(2455,\cdot)\) \(\chi_{2844}(2479,\cdot)\) \(\chi_{2844}(2515,\cdot)\) \(\chi_{2844}(2587,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((1423,317,793)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{25}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 2844 }(1219, a) \) \(1\)\(1\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{49}{78}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{59}{78}\right)\)\(-1\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{67}{78}\right)\)\(e\left(\frac{3}{26}\right)\)
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)
 
\( \chi_{ 2844 }(1219,a) \;\) at \(\;a = \) e.g. 2