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Properties

Label 3038.1219
Modulus $3038$
Conductor $1519$
Order $105$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3038, base_ring=CyclotomicField(210)) M = H._module chi = DirichletCharacter(H, M([30,98]))
 
Copy content gp:[g,chi] = znchar(Mod(1219, 3038))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3038.1219");
 

Basic properties

Modulus: \(3038\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1519\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(105\)
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 \(\chi_{1519}(1219,\cdot)\)
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 3038.cm

\(\chi_{3038}(71,\cdot)\) \(\chi_{3038}(113,\cdot)\) \(\chi_{3038}(169,\cdot)\) \(\chi_{3038}(183,\cdot)\) \(\chi_{3038}(267,\cdot)\) \(\chi_{3038}(351,\cdot)\) \(\chi_{3038}(379,\cdot)\) \(\chi_{3038}(421,\cdot)\) \(\chi_{3038}(505,\cdot)\) \(\chi_{3038}(547,\cdot)\) \(\chi_{3038}(603,\cdot)\) \(\chi_{3038}(617,\cdot)\) \(\chi_{3038}(701,\cdot)\) \(\chi_{3038}(813,\cdot)\) \(\chi_{3038}(855,\cdot)\) \(\chi_{3038}(939,\cdot)\) \(\chi_{3038}(1037,\cdot)\) \(\chi_{3038}(1051,\cdot)\) \(\chi_{3038}(1135,\cdot)\) \(\chi_{3038}(1219,\cdot)\) \(\chi_{3038}(1247,\cdot)\) \(\chi_{3038}(1289,\cdot)\) \(\chi_{3038}(1415,\cdot)\) \(\chi_{3038}(1485,\cdot)\) \(\chi_{3038}(1653,\cdot)\) \(\chi_{3038}(1681,\cdot)\) \(\chi_{3038}(1723,\cdot)\) \(\chi_{3038}(1807,\cdot)\) \(\chi_{3038}(1849,\cdot)\) \(\chi_{3038}(1905,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((1179,1863)\) → \((e\left(\frac{1}{7}\right),e\left(\frac{7}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 3038 }(1219, a) \) \(1\)\(1\)\(e\left(\frac{64}{105}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{23}{105}\right)\)\(e\left(\frac{47}{105}\right)\)\(e\left(\frac{89}{105}\right)\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{88}{105}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{20}{21}\right)\)
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)
 
\( \chi_{ 3038 }(1219,a) \;\) at \(\;a = \) e.g. 2