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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1519, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([150,14]))
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gp:[g,chi] = znchar(Mod(505, 1519))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1519.505");
| Modulus: | \(1519\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1519\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(105\) |
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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;
|
| 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);
|
| 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);
|
\(\chi_{1519}(71,\cdot)\)
\(\chi_{1519}(113,\cdot)\)
\(\chi_{1519}(134,\cdot)\)
\(\chi_{1519}(162,\cdot)\)
\(\chi_{1519}(169,\cdot)\)
\(\chi_{1519}(183,\cdot)\)
\(\chi_{1519}(204,\cdot)\)
\(\chi_{1519}(267,\cdot)\)
\(\chi_{1519}(288,\cdot)\)
\(\chi_{1519}(330,\cdot)\)
\(\chi_{1519}(351,\cdot)\)
\(\chi_{1519}(379,\cdot)\)
\(\chi_{1519}(386,\cdot)\)
\(\chi_{1519}(400,\cdot)\)
\(\chi_{1519}(421,\cdot)\)
\(\chi_{1519}(484,\cdot)\)
\(\chi_{1519}(505,\cdot)\)
\(\chi_{1519}(547,\cdot)\)
\(\chi_{1519}(568,\cdot)\)
\(\chi_{1519}(596,\cdot)\)
\(\chi_{1519}(603,\cdot)\)
\(\chi_{1519}(617,\cdot)\)
\(\chi_{1519}(701,\cdot)\)
\(\chi_{1519}(722,\cdot)\)
\(\chi_{1519}(764,\cdot)\)
\(\chi_{1519}(813,\cdot)\)
\(\chi_{1519}(820,\cdot)\)
\(\chi_{1519}(855,\cdot)\)
\(\chi_{1519}(918,\cdot)\)
\(\chi_{1519}(939,\cdot)\)
...
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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 };
\((1179,344)\) → \((e\left(\frac{5}{7}\right),e\left(\frac{1}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1519 }(505, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{13}{105}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
