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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1647, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([70,27]))
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gp:[g,chi] = znchar(Mod(695, 1647))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1647.695");
| Modulus: | \(1647\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1647\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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);
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\(\chi_{1647}(23,\cdot)\)
\(\chi_{1647}(38,\cdot)\)
\(\chi_{1647}(146,\cdot)\)
\(\chi_{1647}(155,\cdot)\)
\(\chi_{1647}(191,\cdot)\)
\(\chi_{1647}(221,\cdot)\)
\(\chi_{1647}(236,\cdot)\)
\(\chi_{1647}(272,\cdot)\)
\(\chi_{1647}(281,\cdot)\)
\(\chi_{1647}(329,\cdot)\)
\(\chi_{1647}(338,\cdot)\)
\(\chi_{1647}(374,\cdot)\)
\(\chi_{1647}(389,\cdot)\)
\(\chi_{1647}(419,\cdot)\)
\(\chi_{1647}(455,\cdot)\)
\(\chi_{1647}(464,\cdot)\)
\(\chi_{1647}(572,\cdot)\)
\(\chi_{1647}(587,\cdot)\)
\(\chi_{1647}(695,\cdot)\)
\(\chi_{1647}(704,\cdot)\)
\(\chi_{1647}(740,\cdot)\)
\(\chi_{1647}(770,\cdot)\)
\(\chi_{1647}(785,\cdot)\)
\(\chi_{1647}(821,\cdot)\)
\(\chi_{1647}(830,\cdot)\)
\(\chi_{1647}(878,\cdot)\)
\(\chi_{1647}(887,\cdot)\)
\(\chi_{1647}(923,\cdot)\)
\(\chi_{1647}(938,\cdot)\)
\(\chi_{1647}(968,\cdot)\)
...
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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 };
\((245,1405)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{3}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1647 }(695, a) \) |
\(1\) | \(1\) | \(e\left(\frac{97}{180}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{103}{180}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{45}\right)\) |
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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)
