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Properties

Label 90944.1587
Modulus $90944$
Conductor $12992$
Order $336$
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(90944, base_ring=CyclotomicField(336)) M = H._module chi = DirichletCharacter(H, M([168,315,280,204]))
 
Copy content gp:[g,chi] = znchar(Mod(1587, 90944))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90944.1587");
 

Basic properties

Modulus: \(90944\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12992\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(336\)
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 \(\chi_{12992}(1587,\cdot)\)
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 90944.bxq

\(\chi_{90944}(19,\cdot)\) \(\chi_{90944}(619,\cdot)\) \(\chi_{90944}(1587,\cdot)\) \(\chi_{90944}(2763,\cdot)\) \(\chi_{90944}(4331,\cdot)\) \(\chi_{90944}(4539,\cdot)\) \(\chi_{90944}(5715,\cdot)\) \(\chi_{90944}(6499,\cdot)\) \(\chi_{90944}(6891,\cdot)\) \(\chi_{90944}(8251,\cdot)\) \(\chi_{90944}(9427,\cdot)\) \(\chi_{90944}(10211,\cdot)\) \(\chi_{90944}(10603,\cdot)\) \(\chi_{90944}(12771,\cdot)\) \(\chi_{90944}(13163,\cdot)\) \(\chi_{90944}(13947,\cdot)\) \(\chi_{90944}(15123,\cdot)\) \(\chi_{90944}(16483,\cdot)\) \(\chi_{90944}(16875,\cdot)\) \(\chi_{90944}(17659,\cdot)\) \(\chi_{90944}(18835,\cdot)\) \(\chi_{90944}(19043,\cdot)\) \(\chi_{90944}(20611,\cdot)\) \(\chi_{90944}(21787,\cdot)\) \(\chi_{90944}(22755,\cdot)\) \(\chi_{90944}(23355,\cdot)\) \(\chi_{90944}(24323,\cdot)\) \(\chi_{90944}(25499,\cdot)\) \(\chi_{90944}(27067,\cdot)\) \(\chi_{90944}(27275,\cdot)\) ...

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: $\Q(\zeta_{336})$
Fixed field: Number field defined by a degree 336 polynomial (not computed)

Values on generators

\((76735,28421,29697,68993)\) → \((-1,e\left(\frac{15}{16}\right),e\left(\frac{5}{6}\right),e\left(\frac{17}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 90944 }(1587, a) \) \(-1\)\(1\)\(e\left(\frac{61}{336}\right)\)\(e\left(\frac{155}{336}\right)\)\(e\left(\frac{61}{168}\right)\)\(e\left(\frac{235}{336}\right)\)\(e\left(\frac{55}{112}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{233}{336}\right)\)\(e\left(\frac{73}{168}\right)\)\(e\left(\frac{155}{168}\right)\)
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)
 
\( \chi_{ 90944 }(1587,a) \;\) at \(\;a = \) e.g. 2