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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1899, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([175,137]))
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gp:[g,chi] = znchar(Mod(1004, 1899))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1899.1004");
| Modulus: | \(1899\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1899\) |
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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;
|
| 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);
|
\(\chi_{1899}(2,\cdot)\)
\(\chi_{1899}(41,\cdot)\)
\(\chi_{1899}(149,\cdot)\)
\(\chi_{1899}(155,\cdot)\)
\(\chi_{1899}(164,\cdot)\)
\(\chi_{1899}(167,\cdot)\)
\(\chi_{1899}(218,\cdot)\)
\(\chi_{1899}(329,\cdot)\)
\(\chi_{1899}(353,\cdot)\)
\(\chi_{1899}(356,\cdot)\)
\(\chi_{1899}(392,\cdot)\)
\(\chi_{1899}(398,\cdot)\)
\(\chi_{1899}(425,\cdot)\)
\(\chi_{1899}(461,\cdot)\)
\(\chi_{1899}(497,\cdot)\)
\(\chi_{1899}(563,\cdot)\)
\(\chi_{1899}(581,\cdot)\)
\(\chi_{1899}(587,\cdot)\)
\(\chi_{1899}(596,\cdot)\)
\(\chi_{1899}(650,\cdot)\)
\(\chi_{1899}(662,\cdot)\)
\(\chi_{1899}(668,\cdot)\)
\(\chi_{1899}(725,\cdot)\)
\(\chi_{1899}(749,\cdot)\)
\(\chi_{1899}(785,\cdot)\)
\(\chi_{1899}(824,\cdot)\)
\(\chi_{1899}(866,\cdot)\)
\(\chi_{1899}(929,\cdot)\)
\(\chi_{1899}(950,\cdot)\)
\(\chi_{1899}(974,\cdot)\)
...
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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 };
\((1478,424)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{137}{210}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1899 }(1004, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(-1\) | \(e\left(\frac{33}{35}\right)\) |
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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)
