↑

Properties

Label 4136.315
Modulus $4136$
Conductor $4136$
Order $230$
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(4136, base_ring=CyclotomicField(230)) M = H._module chi = DirichletCharacter(H, M([115,115,161,135]))
 
Copy content gp:[g,chi] = znchar(Mod(315, 4136))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4136.315");
 

Basic properties

Modulus: \(4136\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4136\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(230\)
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 4136.cl

\(\chi_{4136}(19,\cdot)\) \(\chi_{4136}(35,\cdot)\) \(\chi_{4136}(107,\cdot)\) \(\chi_{4136}(123,\cdot)\) \(\chi_{4136}(139,\cdot)\) \(\chi_{4136}(171,\cdot)\) \(\chi_{4136}(211,\cdot)\) \(\chi_{4136}(227,\cdot)\) \(\chi_{4136}(315,\cdot)\) \(\chi_{4136}(387,\cdot)\) \(\chi_{4136}(475,\cdot)\) \(\chi_{4136}(547,\cdot)\) \(\chi_{4136}(579,\cdot)\) \(\chi_{4136}(651,\cdot)\) \(\chi_{4136}(699,\cdot)\) \(\chi_{4136}(787,\cdot)\) \(\chi_{4136}(843,\cdot)\) \(\chi_{4136}(875,\cdot)\) \(\chi_{4136}(915,\cdot)\) \(\chi_{4136}(931,\cdot)\) \(\chi_{4136}(963,\cdot)\) \(\chi_{4136}(1075,\cdot)\) \(\chi_{4136}(1091,\cdot)\) \(\chi_{4136}(1107,\cdot)\) \(\chi_{4136}(1139,\cdot)\) \(\chi_{4136}(1163,\cdot)\) \(\chi_{4136}(1195,\cdot)\) \(\chi_{4136}(1227,\cdot)\) \(\chi_{4136}(1251,\cdot)\) \(\chi_{4136}(1267,\cdot)\) ...

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: $\Q(\zeta_{115})$
Fixed field: Number field defined by a degree 230 polynomial (not computed)

Values on generators

\((3103,2069,2257,1321)\) → \((-1,-1,e\left(\frac{7}{10}\right),e\left(\frac{27}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 4136 }(315, a) \) \(-1\)\(1\)\(e\left(\frac{39}{115}\right)\)\(e\left(\frac{102}{115}\right)\)\(e\left(\frac{21}{115}\right)\)\(e\left(\frac{78}{115}\right)\)\(e\left(\frac{151}{230}\right)\)\(e\left(\frac{26}{115}\right)\)\(e\left(\frac{159}{230}\right)\)\(e\left(\frac{59}{115}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{10}{23}\right)\)
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)
 
\( \chi_{ 4136 }(315,a) \;\) at \(\;a = \) e.g. 2