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Properties

Label 8330.2031
Modulus $8330$
Conductor $833$
Order $56$
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(8330, base_ring=CyclotomicField(56)) M = H._module chi = DirichletCharacter(H, M([0,32,35]))
 
Copy content gp:[g,chi] = znchar(Mod(2031, 8330))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8330.2031");
 

Basic properties

Modulus: \(8330\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(833\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(56\)
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 \(\chi_{833}(365,\cdot)\)
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 8330.ed

\(\chi_{8330}(281,\cdot)\) \(\chi_{8330}(631,\cdot)\) \(\chi_{8330}(841,\cdot)\) \(\chi_{8330}(1681,\cdot)\) \(\chi_{8330}(1821,\cdot)\) \(\chi_{8330}(2031,\cdot)\) \(\chi_{8330}(2661,\cdot)\) \(\chi_{8330}(2871,\cdot)\) \(\chi_{8330}(3011,\cdot)\) \(\chi_{8330}(3221,\cdot)\) \(\chi_{8330}(3851,\cdot)\) \(\chi_{8330}(4061,\cdot)\) \(\chi_{8330}(4201,\cdot)\) \(\chi_{8330}(5041,\cdot)\) \(\chi_{8330}(5251,\cdot)\) \(\chi_{8330}(5601,\cdot)\) \(\chi_{8330}(6231,\cdot)\) \(\chi_{8330}(6441,\cdot)\) \(\chi_{8330}(6581,\cdot)\) \(\chi_{8330}(6791,\cdot)\) \(\chi_{8330}(7421,\cdot)\) \(\chi_{8330}(7631,\cdot)\) \(\chi_{8330}(7771,\cdot)\) \(\chi_{8330}(7981,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((1667,2551,2451)\) → \((1,e\left(\frac{4}{7}\right),e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(19\)\(23\)\(27\)\(29\)\(31\)\(33\)
\( \chi_{ 8330 }(2031, a) \) \(1\)\(1\)\(e\left(\frac{11}{56}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{13}{56}\right)\)\(e\left(\frac{5}{14}\right)\)\(-i\)\(e\left(\frac{5}{56}\right)\)\(e\left(\frac{33}{56}\right)\)\(e\left(\frac{23}{56}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{7}\right)\)
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)
 
\( \chi_{ 8330 }(2031,a) \;\) at \(\;a = \) e.g. 2