↑

Properties

Label 1180.651
Modulus $1180$
Conductor $236$
Order $58$
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(1180, base_ring=CyclotomicField(58)) M = H._module chi = DirichletCharacter(H, M([29,0,1]))
 
Copy content gp:[g,chi] = znchar(Mod(651, 1180))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1180.651");
 

Basic properties

Modulus: \(1180\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(236\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(58\)
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 \(\chi_{236}(179,\cdot)\)
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 1180.q

\(\chi_{1180}(11,\cdot)\) \(\chi_{1180}(31,\cdot)\) \(\chi_{1180}(91,\cdot)\) \(\chi_{1180}(111,\cdot)\) \(\chi_{1180}(131,\cdot)\) \(\chi_{1180}(151,\cdot)\) \(\chi_{1180}(191,\cdot)\) \(\chi_{1180}(211,\cdot)\) \(\chi_{1180}(231,\cdot)\) \(\chi_{1180}(291,\cdot)\) \(\chi_{1180}(351,\cdot)\) \(\chi_{1180}(391,\cdot)\) \(\chi_{1180}(431,\cdot)\) \(\chi_{1180}(451,\cdot)\) \(\chi_{1180}(511,\cdot)\) \(\chi_{1180}(571,\cdot)\) \(\chi_{1180}(651,\cdot)\) \(\chi_{1180}(691,\cdot)\) \(\chi_{1180}(731,\cdot)\) \(\chi_{1180}(751,\cdot)\) \(\chi_{1180}(791,\cdot)\) \(\chi_{1180}(811,\cdot)\) \(\chi_{1180}(891,\cdot)\) \(\chi_{1180}(991,\cdot)\) \(\chi_{1180}(1011,\cdot)\) \(\chi_{1180}(1131,\cdot)\) \(\chi_{1180}(1151,\cdot)\) \(\chi_{1180}(1171,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\((591,237,61)\) → \((-1,1,e\left(\frac{1}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1180 }(651, a) \) \(1\)\(1\)\(e\left(\frac{21}{58}\right)\)\(e\left(\frac{47}{58}\right)\)\(e\left(\frac{21}{29}\right)\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{45}{58}\right)\)\(e\left(\frac{20}{29}\right)\)\(e\left(\frac{9}{58}\right)\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{22}{29}\right)\)\(e\left(\frac{5}{58}\right)\)
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)
 
\( \chi_{ 1180 }(651,a) \;\) at \(\;a = \) e.g. 2