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Properties

Label 8820.7409
Modulus $8820$
Conductor $2205$
Order $42$
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(8820, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([0,7,21,13]))
 
Copy content gp:[g,chi] = znchar(Mod(7409, 8820))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8820.7409");
 

Basic properties

Modulus: \(8820\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2205\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(42\)
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 \(\chi_{2205}(794,\cdot)\)
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 8820.go

\(\chi_{8820}(1769,\cdot)\) \(\chi_{8820}(2369,\cdot)\) \(\chi_{8820}(3029,\cdot)\) \(\chi_{8820}(3629,\cdot)\) \(\chi_{8820}(4289,\cdot)\) \(\chi_{8820}(4889,\cdot)\) \(\chi_{8820}(5549,\cdot)\) \(\chi_{8820}(6149,\cdot)\) \(\chi_{8820}(6809,\cdot)\) \(\chi_{8820}(7409,\cdot)\) \(\chi_{8820}(8069,\cdot)\) \(\chi_{8820}(8669,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.42.64499777946714835177141019992254259402911208109553981749879955329445342864388015575823925406164646148681640625.1

Values on generators

\((4411,7841,7057,1081)\) → \((1,e\left(\frac{1}{6}\right),-1,e\left(\frac{13}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8820 }(7409, a) \) \(1\)\(1\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(-1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{1}{42}\right)\)
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)
 
\( \chi_{ 8820 }(7409,a) \;\) at \(\;a = \) e.g. 2