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Properties

Label 35072.581
Modulus $35072$
Conductor $35072$
Order $1088$
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(35072, base_ring=CyclotomicField(1088)) M = H._module chi = DirichletCharacter(H, M([0,833,984]))
 
Copy content gp:[g,chi] = znchar(Mod(581, 35072))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35072.581");
 

Basic properties

Modulus: \(35072\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(35072\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1088\)
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 35072.ha

\(\chi_{35072}(5,\cdot)\) \(\chi_{35072}(45,\cdot)\) \(\chi_{35072}(125,\cdot)\) \(\chi_{35072}(245,\cdot)\) \(\chi_{35072}(317,\cdot)\) \(\chi_{35072}(405,\cdot)\) \(\chi_{35072}(501,\cdot)\) \(\chi_{35072}(581,\cdot)\) \(\chi_{35072}(733,\cdot)\) \(\chi_{35072}(789,\cdot)\) \(\chi_{35072}(845,\cdot)\) \(\chi_{35072}(869,\cdot)\) \(\chi_{35072}(877,\cdot)\) \(\chi_{35072}(893,\cdot)\) \(\chi_{35072}(965,\cdot)\) \(\chi_{35072}(1125,\cdot)\) \(\chi_{35072}(1181,\cdot)\) \(\chi_{35072}(1357,\cdot)\) \(\chi_{35072}(1365,\cdot)\) \(\chi_{35072}(1373,\cdot)\) \(\chi_{35072}(1549,\cdot)\)

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: $\Q(\zeta_{1088})$
Fixed field: Number field defined by a degree 1088 polynomial (not computed)

Values on generators

\((19455,31237,19457)\) → \((1,e\left(\frac{49}{64}\right),e\left(\frac{123}{136}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 35072 }(581, a) \) \(-1\)\(1\)\(e\left(\frac{763}{1088}\right)\)\(e\left(\frac{649}{1088}\right)\)\(e\left(\frac{349}{544}\right)\)\(e\left(\frac{219}{544}\right)\)\(e\left(\frac{453}{1088}\right)\)\(e\left(\frac{647}{1088}\right)\)\(e\left(\frac{81}{272}\right)\)\(e\left(\frac{219}{272}\right)\)\(e\left(\frac{231}{1088}\right)\)\(e\left(\frac{373}{1088}\right)\)
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)
 
\( \chi_{ 35072 }(581,a) \;\) at \(\;a = \) e.g. 2