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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([21,80]))
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gp:[g,chi] = znchar(Mod(1054, 1225))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1225.1054");
| Modulus: | \(1225\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1225\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
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| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
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\(\chi_{1225}(4,\cdot)\)
\(\chi_{1225}(9,\cdot)\)
\(\chi_{1225}(39,\cdot)\)
\(\chi_{1225}(44,\cdot)\)
\(\chi_{1225}(109,\cdot)\)
\(\chi_{1225}(114,\cdot)\)
\(\chi_{1225}(144,\cdot)\)
\(\chi_{1225}(179,\cdot)\)
\(\chi_{1225}(184,\cdot)\)
\(\chi_{1225}(219,\cdot)\)
\(\chi_{1225}(254,\cdot)\)
\(\chi_{1225}(284,\cdot)\)
\(\chi_{1225}(289,\cdot)\)
\(\chi_{1225}(319,\cdot)\)
\(\chi_{1225}(354,\cdot)\)
\(\chi_{1225}(359,\cdot)\)
\(\chi_{1225}(389,\cdot)\)
\(\chi_{1225}(394,\cdot)\)
\(\chi_{1225}(429,\cdot)\)
\(\chi_{1225}(464,\cdot)\)
\(\chi_{1225}(494,\cdot)\)
\(\chi_{1225}(529,\cdot)\)
\(\chi_{1225}(534,\cdot)\)
\(\chi_{1225}(564,\cdot)\)
\(\chi_{1225}(604,\cdot)\)
\(\chi_{1225}(634,\cdot)\)
\(\chi_{1225}(639,\cdot)\)
\(\chi_{1225}(669,\cdot)\)
\(\chi_{1225}(709,\cdot)\)
\(\chi_{1225}(739,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
| Field of values: |
$\Q(\zeta_{105})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 210 polynomial (not computed) |
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sage:chi.fixed_field()
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\((1177,101)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{8}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(1054, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{2}{105}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
