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Properties

Label 9959.359
Modulus $9959$
Conductor $9959$
Order $396$
Real no
Primitive yes
Minimal yes
Parity odd

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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(9959, base_ring=CyclotomicField(396)) M = H._module chi = DirichletCharacter(H, M([378,275]))
 
Copy content gp:[g,chi] = znchar(Mod(359, 9959))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9959.359");
 

Basic properties

Modulus: \(9959\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(9959\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(396\)
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 9959.ck

\(\chi_{9959}(74,\cdot)\) \(\chi_{9959}(158,\cdot)\) \(\chi_{9959}(359,\cdot)\) \(\chi_{9959}(429,\cdot)\) \(\chi_{9959}(503,\cdot)\) \(\chi_{9959}(792,\cdot)\) \(\chi_{9959}(796,\cdot)\) \(\chi_{9959}(862,\cdot)\) \(\chi_{9959}(870,\cdot)\) \(\chi_{9959}(940,\cdot)\) \(\chi_{9959}(1033,\cdot)\) \(\chi_{9959}(1132,\cdot)\) \(\chi_{9959}(1141,\cdot)\) \(\chi_{9959}(1229,\cdot)\) \(\chi_{9959}(1295,\cdot)\) \(\chi_{9959}(1303,\cdot)\) \(\chi_{9959}(1466,\cdot)\) \(\chi_{9959}(1574,\cdot)\) \(\chi_{9959}(1624,\cdot)\) \(\chi_{9959}(1736,\cdot)\) \(\chi_{9959}(1998,\cdot)\) \(\chi_{9959}(2057,\cdot)\) \(\chi_{9959}(2091,\cdot)\) \(\chi_{9959}(2169,\cdot)\) \(\chi_{9959}(2273,\cdot)\) \(\chi_{9959}(2524,\cdot)\) \(\chi_{9959}(2528,\cdot)\) \(\chi_{9959}(2706,\cdot)\) \(\chi_{9959}(2756,\cdot)\) \(\chi_{9959}(2765,\cdot)\) ...

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: $\Q(\zeta_{396})$
Fixed field: Number field defined by a degree 396 polynomial (not computed)

Values on generators

\((9527,438)\) → \((e\left(\frac{21}{22}\right),e\left(\frac{25}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 9959 }(359, a) \) \(-1\)\(1\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{5}{99}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{257}{396}\right)\)\(e\left(\frac{25}{198}\right)\)\(e\left(\frac{241}{396}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{10}{99}\right)\)\(e\left(\frac{287}{396}\right)\)\(e\left(\frac{86}{99}\right)\)
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)
 
\( \chi_{ 9959 }(359,a) \;\) at \(\;a = \) e.g. 2