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Properties

Label 277984.57731
Modulus $277984$
Conductor $277984$
Order $72$
Real no
Primitive yes
Minimal yes
Parity odd

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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(277984, base_ring=CyclotomicField(72)) M = H._module chi = DirichletCharacter(H, M([36,27,24,36,58]))
 
Copy content gp:[g,chi] = znchar(Mod(57731, 277984))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("277984.57731");
 

Basic properties

Modulus: \(277984\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(277984\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(72\)
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 277984.hkw

\(\chi_{277984}(67,\cdot)\) \(\chi_{277984}(19107,\cdot)\) \(\chi_{277984}(22507,\cdot)\) \(\chi_{277984}(31075,\cdot)\) \(\chi_{277984}(54331,\cdot)\) \(\chi_{277984}(57731,\cdot)\) \(\chi_{277984}(73371,\cdot)\) \(\chi_{277984}(86699,\cdot)\) \(\chi_{277984}(95811,\cdot)\) \(\chi_{277984}(96763,\cdot)\) \(\chi_{277984}(125731,\cdot)\) \(\chi_{277984}(134843,\cdot)\) \(\chi_{277984}(139059,\cdot)\) \(\chi_{277984}(158099,\cdot)\) \(\chi_{277984}(161499,\cdot)\) \(\chi_{277984}(170067,\cdot)\) \(\chi_{277984}(193323,\cdot)\) \(\chi_{277984}(196723,\cdot)\) \(\chi_{277984}(212363,\cdot)\) \(\chi_{277984}(225691,\cdot)\) \(\chi_{277984}(234803,\cdot)\) \(\chi_{277984}(235755,\cdot)\) \(\chi_{277984}(264723,\cdot)\) \(\chi_{277984}(273835,\cdot)\)

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: $\Q(\zeta_{72})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 72 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((17375,243237,79425,261633,186593)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{1}{3}\right),-1,e\left(\frac{29}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 277984 }(57731, a) \) \(-1\)\(1\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{25}{72}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{37}{72}\right)\)\(e\left(\frac{11}{72}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{53}{72}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{7}{8}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 277984 }(57731,a) \;\) at \(\;a = \) e.g. 2