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Properties

Label 259200.5977
Modulus $259200$
Conductor $43200$
Order $720$
Real no
Primitive no
Minimal no
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(259200, base_ring=CyclotomicField(720)) M = H._module chi = DirichletCharacter(H, M([0,405,80,36]))
 
Copy content gp:[g,chi] = znchar(Mod(5977, 259200))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("259200.5977");
 

Basic properties

Modulus: \(259200\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(43200\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(720\)
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 \(\chi_{43200}(13477,\cdot)\)
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: no
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 259200.xx

\(\chi_{259200}(73,\cdot)\) \(\chi_{259200}(2953,\cdot)\) \(\chi_{259200}(3097,\cdot)\) \(\chi_{259200}(5977,\cdot)\) \(\chi_{259200}(7273,\cdot)\) \(\chi_{259200}(7417,\cdot)\) \(\chi_{259200}(8713,\cdot)\) \(\chi_{259200}(10297,\cdot)\) \(\chi_{259200}(11737,\cdot)\) \(\chi_{259200}(13033,\cdot)\) \(\chi_{259200}(14617,\cdot)\) \(\chi_{259200}(15913,\cdot)\) \(\chi_{259200}(17353,\cdot)\) \(\chi_{259200}(18937,\cdot)\) \(\chi_{259200}(20233,\cdot)\) \(\chi_{259200}(20377,\cdot)\) \(\chi_{259200}(21673,\cdot)\) \(\chi_{259200}(24553,\cdot)\) \(\chi_{259200}(24697,\cdot)\) \(\chi_{259200}(27577,\cdot)\) \(\chi_{259200}(28873,\cdot)\) \(\chi_{259200}(29017,\cdot)\) \(\chi_{259200}(30313,\cdot)\) \(\chi_{259200}(31897,\cdot)\) \(\chi_{259200}(33337,\cdot)\) \(\chi_{259200}(34633,\cdot)\) \(\chi_{259200}(36217,\cdot)\) \(\chi_{259200}(37513,\cdot)\) \(\chi_{259200}(38953,\cdot)\) \(\chi_{259200}(40537,\cdot)\) ...

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: $\Q(\zeta_{720})$
Fixed field: Number field defined by a degree 720 polynomial (not computed)

Values on generators

\((157951,202501,6401,72577)\) → \((1,e\left(\frac{9}{16}\right),e\left(\frac{1}{9}\right),e\left(\frac{1}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 259200 }(5977, a) \) \(-1\)\(1\)\(e\left(\frac{47}{72}\right)\)\(e\left(\frac{41}{720}\right)\)\(e\left(\frac{199}{720}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{41}{240}\right)\)\(e\left(\frac{233}{360}\right)\)\(e\left(\frac{287}{720}\right)\)\(e\left(\frac{11}{90}\right)\)\(e\left(\frac{43}{240}\right)\)\(e\left(\frac{347}{360}\right)\)
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)
 
\( \chi_{ 259200 }(5977,a) \;\) at \(\;a = \) e.g. 2