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Properties

Label 12544.323
Modulus $12544$
Conductor $12544$
Order $448$
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(12544, base_ring=CyclotomicField(448)) M = H._module chi = DirichletCharacter(H, M([224,357,192]))
 
Copy content gp:[g,chi] = znchar(Mod(323, 12544))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12544.323");
 

Basic properties

Modulus: \(12544\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12544\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(448\)
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 12544.dw

\(\chi_{12544}(43,\cdot)\) \(\chi_{12544}(155,\cdot)\) \(\chi_{12544}(211,\cdot)\) \(\chi_{12544}(267,\cdot)\) \(\chi_{12544}(323,\cdot)\) \(\chi_{12544}(379,\cdot)\) \(\chi_{12544}(435,\cdot)\) \(\chi_{12544}(547,\cdot)\) \(\chi_{12544}(603,\cdot)\) \(\chi_{12544}(659,\cdot)\) \(\chi_{12544}(715,\cdot)\) \(\chi_{12544}(771,\cdot)\) \(\chi_{12544}(827,\cdot)\) \(\chi_{12544}(939,\cdot)\) \(\chi_{12544}(995,\cdot)\) \(\chi_{12544}(1051,\cdot)\) \(\chi_{12544}(1107,\cdot)\) \(\chi_{12544}(1163,\cdot)\) \(\chi_{12544}(1219,\cdot)\) \(\chi_{12544}(1331,\cdot)\) \(\chi_{12544}(1387,\cdot)\) \(\chi_{12544}(1443,\cdot)\) \(\chi_{12544}(1499,\cdot)\) \(\chi_{12544}(1555,\cdot)\) \(\chi_{12544}(1611,\cdot)\) \(\chi_{12544}(1723,\cdot)\) \(\chi_{12544}(1779,\cdot)\) \(\chi_{12544}(1835,\cdot)\) \(\chi_{12544}(1891,\cdot)\) \(\chi_{12544}(1947,\cdot)\) ...

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: $\Q(\zeta_{448})$
Fixed field: Number field defined by a degree 448 polynomial (not computed)

Values on generators

\((4607,3333,4609)\) → \((-1,e\left(\frac{51}{64}\right),e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 12544 }(323, a) \) \(-1\)\(1\)\(e\left(\frac{367}{448}\right)\)\(e\left(\frac{101}{448}\right)\)\(e\left(\frac{143}{224}\right)\)\(e\left(\frac{169}{448}\right)\)\(e\left(\frac{267}{448}\right)\)\(e\left(\frac{5}{112}\right)\)\(e\left(\frac{3}{112}\right)\)\(e\left(\frac{53}{64}\right)\)\(e\left(\frac{211}{224}\right)\)\(e\left(\frac{101}{224}\right)\)
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)
 
\( \chi_{ 12544 }(323,a) \;\) at \(\;a = \) e.g. 2