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Properties

Label 17550.14641
Modulus $17550$
Conductor $8775$
Order $45$
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(17550, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([80,18,30]))
 
Copy content gp:[g,chi] = znchar(Mod(14641, 17550))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("17550.14641");
 

Basic properties

Modulus: \(17550\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8775\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(45\)
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 \(\chi_{8775}(5866,\cdot)\)
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 17550.ie

\(\chi_{17550}(841,\cdot)\) \(\chi_{17550}(1771,\cdot)\) \(\chi_{17550}(2011,\cdot)\) \(\chi_{17550}(2941,\cdot)\) \(\chi_{17550}(3181,\cdot)\) \(\chi_{17550}(4111,\cdot)\) \(\chi_{17550}(5281,\cdot)\) \(\chi_{17550}(5521,\cdot)\) \(\chi_{17550}(6691,\cdot)\) \(\chi_{17550}(7621,\cdot)\) \(\chi_{17550}(7861,\cdot)\) \(\chi_{17550}(8791,\cdot)\) \(\chi_{17550}(9031,\cdot)\) \(\chi_{17550}(9961,\cdot)\) \(\chi_{17550}(11131,\cdot)\) \(\chi_{17550}(11371,\cdot)\) \(\chi_{17550}(12541,\cdot)\) \(\chi_{17550}(13471,\cdot)\) \(\chi_{17550}(13711,\cdot)\) \(\chi_{17550}(14641,\cdot)\) \(\chi_{17550}(14881,\cdot)\) \(\chi_{17550}(15811,\cdot)\) \(\chi_{17550}(16981,\cdot)\) \(\chi_{17550}(17221,\cdot)\)

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: $\Q(\zeta_{45})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 45 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((9101,9127,8101)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{1}{5}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 17550 }(14641, a) \) \(1\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{2}{9}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 17550 }(14641,a) \;\) at \(\;a = \) e.g. 2