↑

Properties

Label 9398.7407
Modulus $9398$
Conductor $4699$
Order $63$
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(9398, base_ring=CyclotomicField(126)) M = H._module chi = DirichletCharacter(H, M([112,80]))
 
Copy content gp:[g,chi] = znchar(Mod(7407, 9398))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9398.7407");
 

Basic properties

Modulus: \(9398\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4699\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(63\)
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 \(\chi_{4699}(2708,\cdot)\)
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 9398.ex

\(\chi_{9398}(145,\cdot)\) \(\chi_{9398}(231,\cdot)\) \(\chi_{9398}(773,\cdot)\) \(\chi_{9398}(793,\cdot)\) \(\chi_{9398}(1033,\cdot)\) \(\chi_{9398}(1385,\cdot)\) \(\chi_{9398}(1459,\cdot)\) \(\chi_{9398}(1859,\cdot)\) \(\chi_{9398}(2047,\cdot)\) \(\chi_{9398}(2229,\cdot)\) \(\chi_{9398}(2787,\cdot)\) \(\chi_{9398}(2791,\cdot)\) \(\chi_{9398}(2957,\cdot)\) \(\chi_{9398}(3709,\cdot)\) \(\chi_{9398}(4621,\cdot)\) \(\chi_{9398}(4733,\cdot)\) \(\chi_{9398}(5115,\cdot)\) \(\chi_{9398}(5159,\cdot)\) \(\chi_{9398}(5455,\cdot)\) \(\chi_{9398}(5657,\cdot)\) \(\chi_{9398}(5855,\cdot)\) \(\chi_{9398}(6297,\cdot)\) \(\chi_{9398}(6371,\cdot)\) \(\chi_{9398}(6519,\cdot)\) \(\chi_{9398}(6815,\cdot)\) \(\chi_{9398}(7407,\cdot)\) \(\chi_{9398}(7523,\cdot)\) \(\chi_{9398}(7581,\cdot)\) \(\chi_{9398}(8099,\cdot)\) \(\chi_{9398}(8137,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 63 polynomial

Values on generators

\((5589,7623)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{40}{63}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 9398 }(7407, a) \) \(1\)\(1\)\(e\left(\frac{47}{63}\right)\)\(e\left(\frac{43}{63}\right)\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{31}{63}\right)\)\(e\left(\frac{53}{63}\right)\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{22}{63}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{13}{63}\right)\)
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)
 
\( \chi_{ 9398 }(7407,a) \;\) at \(\;a = \) e.g. 2