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Properties

Label 40248.5195
Modulus $40248$
Conductor $40248$
Order $84$
Real no
Primitive yes
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(40248, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([42,42,14,21,36]))
 
Copy content gp:[g,chi] = znchar(Mod(5195, 40248))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("40248.5195");
 

Basic properties

Modulus: \(40248\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(40248\)
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: 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 40248.bsb

\(\chi_{40248}(1139,\cdot)\) \(\chi_{40248}(2075,\cdot)\) \(\chi_{40248}(5195,\cdot)\) \(\chi_{40248}(6755,\cdot)\) \(\chi_{40248}(7259,\cdot)\) \(\chi_{40248}(10379,\cdot)\) \(\chi_{40248}(12683,\cdot)\) \(\chi_{40248}(14555,\cdot)\) \(\chi_{40248}(14747,\cdot)\) \(\chi_{40248}(15491,\cdot)\) \(\chi_{40248}(16619,\cdot)\) \(\chi_{40248}(17555,\cdot)\) \(\chi_{40248}(20171,\cdot)\) \(\chi_{40248}(20675,\cdot)\) \(\chi_{40248}(21731,\cdot)\) \(\chi_{40248}(22235,\cdot)\) \(\chi_{40248}(28163,\cdot)\) \(\chi_{40248}(30035,\cdot)\) \(\chi_{40248}(30971,\cdot)\) \(\chi_{40248}(32027,\cdot)\) \(\chi_{40248}(35147,\cdot)\) \(\chi_{40248}(35651,\cdot)\) \(\chi_{40248}(37211,\cdot)\) \(\chi_{40248}(39515,\cdot)\)

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

\((10063,20125,35777,21673,15913)\) → \((-1,-1,e\left(\frac{1}{6}\right),i,e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 40248 }(5195, a) \) \(-1\)\(1\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{65}{84}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{55}{84}\right)\)\(e\left(\frac{3}{14}\right)\)
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)
 
\( \chi_{ 40248 }(5195,a) \;\) at \(\;a = \) e.g. 2