↑

Properties

Label 174845.721
Modulus $174845$
Conductor $34969$
Order $14960$
Real no
Primitive no
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(174845, base_ring=CyclotomicField(14960)) M = H._module chi = DirichletCharacter(H, M([0,2584,12045]))
 
Copy content gp:[g,chi] = znchar(Mod(721, 174845))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("174845.721");
 

Basic properties

Modulus: \(174845\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(34969\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(14960\)
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 \(\chi_{34969}(721,\cdot)\)
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 174845.ky

\(\chi_{174845}(6,\cdot)\) \(\chi_{174845}(41,\cdot)\) \(\chi_{174845}(46,\cdot)\) \(\chi_{174845}(61,\cdot)\) \(\chi_{174845}(96,\cdot)\) \(\chi_{174845}(116,\cdot)\) \(\chi_{174845}(156,\cdot)\) \(\chi_{174845}(211,\cdot)\) \(\chi_{174845}(216,\cdot)\) \(\chi_{174845}(226,\cdot)\) \(\chi_{174845}(261,\cdot)\) \(\chi_{174845}(266,\cdot)\) \(\chi_{174845}(316,\cdot)\) \(\chi_{174845}(326,\cdot)\) \(\chi_{174845}(371,\cdot)\) \(\chi_{174845}(381,\cdot)\) \(\chi_{174845}(431,\cdot)\) \(\chi_{174845}(436,\cdot)\) \(\chi_{174845}(486,\cdot)\) \(\chi_{174845}(541,\cdot)\) \(\chi_{174845}(556,\cdot)\) \(\chi_{174845}(601,\cdot)\) \(\chi_{174845}(651,\cdot)\) \(\chi_{174845}(656,\cdot)\) \(\chi_{174845}(666,\cdot)\) \(\chi_{174845}(711,\cdot)\) \(\chi_{174845}(721,\cdot)\) \(\chi_{174845}(776,\cdot)\) \(\chi_{174845}(811,\cdot)\) \(\chi_{174845}(821,\cdot)\) ...

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: $\Q(\zeta_{14960})$
Fixed field: Number field defined by a degree 14960 polynomial (not computed)

Values on generators

\((139877,99706,45376)\) → \((1,e\left(\frac{19}{110}\right),e\left(\frac{219}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 174845 }(721, a) \) \(1\)\(1\)\(e\left(\frac{1127}{7480}\right)\)\(e\left(\frac{7}{1360}\right)\)\(e\left(\frac{1127}{3740}\right)\)\(e\left(\frac{2331}{14960}\right)\)\(e\left(\frac{103}{14960}\right)\)\(e\left(\frac{3381}{7480}\right)\)\(e\left(\frac{7}{680}\right)\)\(e\left(\frac{917}{2992}\right)\)\(e\left(\frac{951}{3740}\right)\)\(e\left(\frac{2357}{14960}\right)\)
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)
 
\( \chi_{ 174845 }(721,a) \;\) at \(\;a = \) e.g. 2