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Properties

Label 675.56
Modulus $675$
Conductor $675$
Order $90$
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(675, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([5,36]))
 
Copy content gp:[g,chi] = znchar(Mod(56, 675))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.56");
 

Basic properties

Modulus: \(675\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(675\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
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 675.bf

\(\chi_{675}(11,\cdot)\) \(\chi_{675}(41,\cdot)\) \(\chi_{675}(56,\cdot)\) \(\chi_{675}(86,\cdot)\) \(\chi_{675}(131,\cdot)\) \(\chi_{675}(146,\cdot)\) \(\chi_{675}(191,\cdot)\) \(\chi_{675}(221,\cdot)\) \(\chi_{675}(236,\cdot)\) \(\chi_{675}(266,\cdot)\) \(\chi_{675}(281,\cdot)\) \(\chi_{675}(311,\cdot)\) \(\chi_{675}(356,\cdot)\) \(\chi_{675}(371,\cdot)\) \(\chi_{675}(416,\cdot)\) \(\chi_{675}(446,\cdot)\) \(\chi_{675}(461,\cdot)\) \(\chi_{675}(491,\cdot)\) \(\chi_{675}(506,\cdot)\) \(\chi_{675}(536,\cdot)\) \(\chi_{675}(581,\cdot)\) \(\chi_{675}(596,\cdot)\) \(\chi_{675}(641,\cdot)\) \(\chi_{675}(671,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((326,352)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 675 }(56, a) \) \(-1\)\(1\)\(e\left(\frac{41}{90}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{11}{90}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{13}{15}\right)\)
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)
 
\( \chi_{ 675 }(56,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content comment:Gauss sum
 
Copy content sage:chi.gauss_sum(a)
 
Copy content gp:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 675 }(56,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 675 }(56,·),\chi_{ 675 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 675 }(56,·)) \;\) at \(\; a,b = \) e.g. 1,2