↑

Properties

Label 46748.31771
Modulus $46748$
Conductor $46748$
Order $70$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(46748, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([35,35,10,7]))
 
Copy content gp:[g,chi] = znchar(Mod(31771, 46748))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("46748.31771");
 

Basic properties

Modulus: \(46748\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(46748\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(70\)
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 46748.tz

\(\chi_{46748}(6967,\cdot)\) \(\chi_{46748}(7331,\cdot)\) \(\chi_{46748}(8579,\cdot)\) \(\chi_{46748}(11127,\cdot)\) \(\chi_{46748}(12167,\cdot)\) \(\chi_{46748}(14351,\cdot)\) \(\chi_{46748}(15651,\cdot)\) \(\chi_{46748}(18875,\cdot)\) \(\chi_{46748}(22411,\cdot)\) \(\chi_{46748}(24699,\cdot)\) \(\chi_{46748}(25063,\cdot)\) \(\chi_{46748}(26675,\cdot)\) \(\chi_{46748}(26935,\cdot)\) \(\chi_{46748}(27247,\cdot)\) \(\chi_{46748}(27923,\cdot)\) \(\chi_{46748}(31771,\cdot)\) \(\chi_{46748}(35983,\cdot)\) \(\chi_{46748}(40143,\cdot)\) \(\chi_{46748}(40819,\cdot)\) \(\chi_{46748}(41755,\cdot)\) \(\chi_{46748}(42795,\cdot)\) \(\chi_{46748}(44667,\cdot)\) \(\chi_{46748}(46019,\cdot)\) \(\chi_{46748}(46279,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((23375,17981,19345,42225)\) → \((-1,-1,e\left(\frac{1}{7}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 46748 }(31771, a) \) \(1\)\(1\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{67}{70}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{29}{35}\right)\)\(e\left(\frac{2}{35}\right)\)
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)
 
\( \chi_{ 46748 }(31771,a) \;\) at \(\;a = \) e.g. 2