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Properties

Label 15575.198
Modulus $15575$
Conductor $15575$
Order $660$
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(15575, base_ring=CyclotomicField(660)) M = H._module chi = DirichletCharacter(H, M([363,220,105]))
 
Copy content gp:[g,chi] = znchar(Mod(198, 15575))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15575.198");
 

Basic properties

Modulus: \(15575\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(15575\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(660\)
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 15575.if

\(\chi_{15575}(53,\cdot)\) \(\chi_{15575}(72,\cdot)\) \(\chi_{15575}(198,\cdot)\) \(\chi_{15575}(247,\cdot)\) \(\chi_{15575}(338,\cdot)\) \(\chi_{15575}(373,\cdot)\) \(\chi_{15575}(403,\cdot)\) \(\chi_{15575}(487,\cdot)\) \(\chi_{15575}(513,\cdot)\) \(\chi_{15575}(702,\cdot)\) \(\chi_{15575}(837,\cdot)\) \(\chi_{15575}(1073,\cdot)\) \(\chi_{15575}(1108,\cdot)\) \(\chi_{15575}(1117,\cdot)\) \(\chi_{15575}(1152,\cdot)\) \(\chi_{15575}(1388,\cdot)\) \(\chi_{15575}(1523,\cdot)\) \(\chi_{15575}(1712,\cdot)\) \(\chi_{15575}(1738,\cdot)\) \(\chi_{15575}(1822,\cdot)\) \(\chi_{15575}(1852,\cdot)\) \(\chi_{15575}(1887,\cdot)\) \(\chi_{15575}(1978,\cdot)\) \(\chi_{15575}(2027,\cdot)\) \(\chi_{15575}(2153,\cdot)\) \(\chi_{15575}(2172,\cdot)\) \(\chi_{15575}(2412,\cdot)\) \(\chi_{15575}(2452,\cdot)\) \(\chi_{15575}(2487,\cdot)\) \(\chi_{15575}(2853,\cdot)\) ...

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: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

Values on generators

\((7477,11126,5076)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{1}{3}\right),e\left(\frac{7}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 15575 }(198, a) \) \(-1\)\(1\)\(e\left(\frac{503}{660}\right)\)\(e\left(\frac{113}{330}\right)\)\(e\left(\frac{173}{330}\right)\)\(e\left(\frac{23}{220}\right)\)\(e\left(\frac{63}{220}\right)\)\(e\left(\frac{113}{165}\right)\)\(e\left(\frac{82}{165}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{6}{55}\right)\)\(e\left(\frac{8}{165}\right)\)
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)
 
\( \chi_{ 15575 }(198,a) \;\) at \(\;a = \) e.g. 2