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Properties

Label 35476.22437
Modulus $35476$
Conductor $8869$
Order $84$
Real no
Primitive no
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(35476, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([0,16,49]))
 
Copy content gp:[g,chi] = znchar(Mod(22437, 35476))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35476.22437");
 

Basic properties

Modulus: \(35476\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8869\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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: no, induced from \(\chi_{8869}(4699,\cdot)\)
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 35476.ih

\(\chi_{35476}(2165,\cdot)\) \(\chi_{35476}(2741,\cdot)\) \(\chi_{35476}(3413,\cdot)\) \(\chi_{35476}(4713,\cdot)\) \(\chi_{35476}(8481,\cdot)\) \(\chi_{35476}(12301,\cdot)\) \(\chi_{35476}(12877,\cdot)\) \(\chi_{35476}(13549,\cdot)\) \(\chi_{35476}(14849,\cdot)\) \(\chi_{35476}(17369,\cdot)\) \(\chi_{35476}(17945,\cdot)\) \(\chi_{35476}(18617,\cdot)\) \(\chi_{35476}(19917,\cdot)\) \(\chi_{35476}(22437,\cdot)\) \(\chi_{35476}(23013,\cdot)\) \(\chi_{35476}(24985,\cdot)\) \(\chi_{35476}(27505,\cdot)\) \(\chi_{35476}(28081,\cdot)\) \(\chi_{35476}(28753,\cdot)\) \(\chi_{35476}(30053,\cdot)\) \(\chi_{35476}(32573,\cdot)\) \(\chi_{35476}(33149,\cdot)\) \(\chi_{35476}(33821,\cdot)\) \(\chi_{35476}(35121,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((17739,22445,33125)\) → \((1,e\left(\frac{4}{21}\right),e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 35476 }(22437, a) \) \(-1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{1}{21}\right)\)
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)
 
\( \chi_{ 35476 }(22437,a) \;\) at \(\;a = \) e.g. 2