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Properties

Label 15341.3081
Modulus $15341$
Conductor $15341$
Order $322$
Real no
Primitive yes
Minimal yes
Parity odd

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(15341, base_ring=CyclotomicField(322)) M = H._module chi = DirichletCharacter(H, M([119,138]))
 
Copy content gp:[g,chi] = znchar(Mod(3081, 15341))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15341.3081");
 

Basic properties

Modulus: \(15341\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(15341\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(322\)
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 15341.bi

\(\chi_{15341}(45,\cdot)\) \(\chi_{15341}(252,\cdot)\) \(\chi_{15341}(344,\cdot)\) \(\chi_{15341}(413,\cdot)\) \(\chi_{15341}(459,\cdot)\) \(\chi_{15341}(574,\cdot)\) \(\chi_{15341}(712,\cdot)\) \(\chi_{15341}(919,\cdot)\) \(\chi_{15341}(1011,\cdot)\) \(\chi_{15341}(1080,\cdot)\) \(\chi_{15341}(1126,\cdot)\) \(\chi_{15341}(1241,\cdot)\) \(\chi_{15341}(1379,\cdot)\) \(\chi_{15341}(1678,\cdot)\) \(\chi_{15341}(1747,\cdot)\) \(\chi_{15341}(1793,\cdot)\) \(\chi_{15341}(1908,\cdot)\) \(\chi_{15341}(2046,\cdot)\) \(\chi_{15341}(2253,\cdot)\) \(\chi_{15341}(2345,\cdot)\) \(\chi_{15341}(2414,\cdot)\) \(\chi_{15341}(2460,\cdot)\) \(\chi_{15341}(2575,\cdot)\) \(\chi_{15341}(2713,\cdot)\) \(\chi_{15341}(2920,\cdot)\) \(\chi_{15341}(3012,\cdot)\) \(\chi_{15341}(3081,\cdot)\) \(\chi_{15341}(3127,\cdot)\) \(\chi_{15341}(3242,\cdot)\) \(\chi_{15341}(3380,\cdot)\) ...

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: $\Q(\zeta_{161})$
Fixed field: Number field defined by a degree 322 polynomial (not computed)

Values on generators

\((8469,13226)\) → \((e\left(\frac{17}{46}\right),e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 15341 }(3081, a) \) \(-1\)\(1\)\(e\left(\frac{55}{161}\right)\)\(e\left(\frac{9}{161}\right)\)\(e\left(\frac{110}{161}\right)\)\(e\left(\frac{257}{322}\right)\)\(e\left(\frac{64}{161}\right)\)\(e\left(\frac{263}{322}\right)\)\(e\left(\frac{4}{161}\right)\)\(e\left(\frac{18}{161}\right)\)\(e\left(\frac{45}{322}\right)\)\(e\left(\frac{209}{322}\right)\)
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)
 
\( \chi_{ 15341 }(3081,a) \;\) at \(\;a = \) e.g. 2