↑

Properties

Label 11515.617
Modulus $11515$
Conductor $11515$
Order $644$
Real no
Primitive yes
Minimal yes
Parity odd

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(11515, base_ring=CyclotomicField(644)) M = H._module chi = DirichletCharacter(H, M([161,276,532]))
 
Copy content gp:[g,chi] = znchar(Mod(617, 11515))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11515.617");
 

Basic properties

Modulus: \(11515\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11515\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(644\)
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 11515.dg

\(\chi_{11515}(8,\cdot)\) \(\chi_{11515}(162,\cdot)\) \(\chi_{11515}(183,\cdot)\) \(\chi_{11515}(253,\cdot)\) \(\chi_{11515}(267,\cdot)\) \(\chi_{11515}(288,\cdot)\) \(\chi_{11515}(337,\cdot)\) \(\chi_{11515}(477,\cdot)\) \(\chi_{11515}(498,\cdot)\) \(\chi_{11515}(512,\cdot)\) \(\chi_{11515}(533,\cdot)\) \(\chi_{11515}(568,\cdot)\) \(\chi_{11515}(582,\cdot)\) \(\chi_{11515}(617,\cdot)\) \(\chi_{11515}(708,\cdot)\) \(\chi_{11515}(722,\cdot)\) \(\chi_{11515}(813,\cdot)\) \(\chi_{11515}(827,\cdot)\) \(\chi_{11515}(848,\cdot)\) \(\chi_{11515}(862,\cdot)\) \(\chi_{11515}(897,\cdot)\) \(\chi_{11515}(918,\cdot)\) \(\chi_{11515}(967,\cdot)\) \(\chi_{11515}(1023,\cdot)\) \(\chi_{11515}(1037,\cdot)\) \(\chi_{11515}(1058,\cdot)\) \(\chi_{11515}(1093,\cdot)\) \(\chi_{11515}(1142,\cdot)\) \(\chi_{11515}(1212,\cdot)\) \(\chi_{11515}(1247,\cdot)\) ...

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: $\Q(\zeta_{644})$
Fixed field: Number field defined by a degree 644 polynomial (not computed)

Values on generators

\((4607,9166,9311)\) → \((i,e\left(\frac{3}{7}\right),e\left(\frac{19}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 11515 }(617, a) \) \(-1\)\(1\)\(e\left(\frac{169}{644}\right)\)\(e\left(\frac{451}{644}\right)\)\(e\left(\frac{169}{322}\right)\)\(e\left(\frac{155}{161}\right)\)\(e\left(\frac{507}{644}\right)\)\(e\left(\frac{129}{322}\right)\)\(e\left(\frac{149}{161}\right)\)\(e\left(\frac{145}{644}\right)\)\(e\left(\frac{631}{644}\right)\)\(e\left(\frac{8}{161}\right)\)
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)
 
\( \chi_{ 11515 }(617,a) \;\) at \(\;a = \) e.g. 2