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Properties

Label 8740.441
Modulus $8740$
Conductor $437$
Order $99$
Real no
Primitive no
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(8740, base_ring=CyclotomicField(198)) M = H._module chi = DirichletCharacter(H, M([0,0,22,36]))
 
Copy content gp:[g,chi] = znchar(Mod(441, 8740))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8740.441");
 

Basic properties

Modulus: \(8740\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(437\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(99\)
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 \(\chi_{437}(4,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 8740.ei

\(\chi_{8740}(81,\cdot)\) \(\chi_{8740}(101,\cdot)\) \(\chi_{8740}(301,\cdot)\) \(\chi_{8740}(441,\cdot)\) \(\chi_{8740}(541,\cdot)\) \(\chi_{8740}(821,\cdot)\) \(\chi_{8740}(841,\cdot)\) \(\chi_{8740}(1061,\cdot)\) \(\chi_{8740}(1221,\cdot)\) \(\chi_{8740}(1301,\cdot)\) \(\chi_{8740}(1461,\cdot)\) \(\chi_{8740}(1681,\cdot)\) \(\chi_{8740}(1821,\cdot)\) \(\chi_{8740}(1961,\cdot)\) \(\chi_{8740}(1981,\cdot)\) \(\chi_{8740}(2201,\cdot)\) \(\chi_{8740}(2221,\cdot)\) \(\chi_{8740}(2341,\cdot)\) \(\chi_{8740}(2381,\cdot)\) \(\chi_{8740}(2441,\cdot)\) \(\chi_{8740}(2601,\cdot)\) \(\chi_{8740}(2741,\cdot)\) \(\chi_{8740}(3121,\cdot)\) \(\chi_{8740}(3141,\cdot)\) \(\chi_{8740}(3201,\cdot)\) \(\chi_{8740}(3341,\cdot)\) \(\chi_{8740}(3361,\cdot)\) \(\chi_{8740}(3481,\cdot)\) \(\chi_{8740}(3521,\cdot)\) \(\chi_{8740}(3581,\cdot)\) ...

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: $\Q(\zeta_{99})$
Fixed field: Number field defined by a degree 99 polynomial

Values on generators

\((4371,3497,2301,3041)\) → \((1,1,e\left(\frac{1}{9}\right),e\left(\frac{2}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(27\)\(29\)\(31\)
\( \chi_{ 8740 }(441, a) \) \(1\)\(1\)\(e\left(\frac{35}{99}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{70}{99}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{10}{99}\right)\)\(e\left(\frac{38}{99}\right)\)\(e\left(\frac{47}{99}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{16}{99}\right)\)\(e\left(\frac{25}{33}\right)\)
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)
 
\( \chi_{ 8740 }(441,a) \;\) at \(\;a = \) e.g. 2