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Properties

Label 404.227
Modulus $404$
Conductor $404$
Order $50$
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(404, base_ring=CyclotomicField(50)) M = H._module chi = DirichletCharacter(H, M([25,24]))
 
Copy content gp:[g,chi] = znchar(Mod(227, 404))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("404.227");
 

Basic properties

Modulus: \(404\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(404\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(50\)
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 404.o

\(\chi_{404}(19,\cdot)\) \(\chi_{404}(31,\cdot)\) \(\chi_{404}(71,\cdot)\) \(\chi_{404}(79,\cdot)\) \(\chi_{404}(155,\cdot)\) \(\chi_{404}(159,\cdot)\) \(\chi_{404}(179,\cdot)\) \(\chi_{404}(207,\cdot)\) \(\chi_{404}(227,\cdot)\) \(\chi_{404}(239,\cdot)\) \(\chi_{404}(283,\cdot)\) \(\chi_{404}(299,\cdot)\) \(\chi_{404}(319,\cdot)\) \(\chi_{404}(327,\cdot)\) \(\chi_{404}(355,\cdot)\) \(\chi_{404}(359,\cdot)\) \(\chi_{404}(371,\cdot)\) \(\chi_{404}(383,\cdot)\) \(\chi_{404}(391,\cdot)\) \(\chi_{404}(395,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((203,305)\) → \((-1,e\left(\frac{12}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 404 }(227, a) \) \(-1\)\(1\)\(e\left(\frac{31}{50}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{41}{50}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{17}{25}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{11}{25}\right)\)
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)
 
\( \chi_{ 404 }(227,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 404 }(227,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 404 }(227,·),\chi_{ 404 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 404 }(227,·)) \;\) at \(\; a,b = \) e.g. 1,2