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Properties

Label 27090.19213
Modulus $27090$
Conductor $13545$
Order $84$
Real no
Primitive no
Minimal yes
Parity even

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(27090, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([56,63,70,36]))
 
Copy content gp:[g,chi] = znchar(Mod(19213, 27090))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27090.19213");
 

Basic properties

Modulus: \(27090\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(13545\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 \(\chi_{13545}(5668,\cdot)\)
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 27090.wf

\(\chi_{27090}(1417,\cdot)\) \(\chi_{27090}(2677,\cdot)\) \(\chi_{27090}(2707,\cdot)\) \(\chi_{27090}(3967,\cdot)\) \(\chi_{27090}(4063,\cdot)\) \(\chi_{27090}(5353,\cdot)\) \(\chi_{27090}(6583,\cdot)\) \(\chi_{27090}(7087,\cdot)\) \(\chi_{27090}(7873,\cdot)\) \(\chi_{27090}(8377,\cdot)\) \(\chi_{27090}(11497,\cdot)\) \(\chi_{27090}(12253,\cdot)\) \(\chi_{27090}(12787,\cdot)\) \(\chi_{27090}(13513,\cdot)\) \(\chi_{27090}(13543,\cdot)\) \(\chi_{27090}(14803,\cdot)\) \(\chi_{27090}(17923,\cdot)\) \(\chi_{27090}(19213,\cdot)\) \(\chi_{27090}(20317,\cdot)\) \(\chi_{27090}(21607,\cdot)\) \(\chi_{27090}(22333,\cdot)\) \(\chi_{27090}(22837,\cdot)\) \(\chi_{27090}(23623,\cdot)\) \(\chi_{27090}(24127,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((21071,10837,19351,25201)\) → \((e\left(\frac{2}{3}\right),-i,e\left(\frac{5}{6}\right),e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(47\)
\( \chi_{ 27090 }(19213, a) \) \(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{61}{84}\right)\)
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)
 
\( \chi_{ 27090 }(19213,a) \;\) at \(\;a = \) e.g. 2