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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24037, base_ring=CyclotomicField(3612))
M = H._module
chi = DirichletCharacter(H, M([301,440]))
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gp:[g,chi] = znchar(Mod(483, 24037))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("24037.483");
| Modulus: | \(24037\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(24037\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(3612\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{24037}(24,\cdot)\)
\(\chi_{24037}(58,\cdot)\)
\(\chi_{24037}(67,\cdot)\)
\(\chi_{24037}(111,\cdot)\)
\(\chi_{24037}(167,\cdot)\)
\(\chi_{24037}(189,\cdot)\)
\(\chi_{24037}(228,\cdot)\)
\(\chi_{24037}(240,\cdot)\)
\(\chi_{24037}(267,\cdot)\)
\(\chi_{24037}(318,\cdot)\)
\(\chi_{24037}(357,\cdot)\)
\(\chi_{24037}(358,\cdot)\)
\(\chi_{24037}(375,\cdot)\)
\(\chi_{24037}(396,\cdot)\)
\(\chi_{24037}(418,\cdot)\)
\(\chi_{24037}(427,\cdot)\)
\(\chi_{24037}(440,\cdot)\)
\(\chi_{24037}(453,\cdot)\)
\(\chi_{24037}(470,\cdot)\)
\(\chi_{24037}(483,\cdot)\)
\(\chi_{24037}(487,\cdot)\)
\(\chi_{24037}(488,\cdot)\)
\(\chi_{24037}(496,\cdot)\)
\(\chi_{24037}(583,\cdot)\)
\(\chi_{24037}(617,\cdot)\)
\(\chi_{24037}(626,\cdot)\)
\(\chi_{24037}(670,\cdot)\)
\(\chi_{24037}(726,\cdot)\)
\(\chi_{24037}(748,\cdot)\)
\(\chi_{24037}(769,\cdot)\)
...
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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 };
\((16642,14795)\) → \((e\left(\frac{1}{12}\right),e\left(\frac{110}{903}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 24037 }(483, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{2017}{3612}\right)\) | \(e\left(\frac{137}{301}\right)\) | \(e\left(\frac{211}{1806}\right)\) | \(e\left(\frac{941}{3612}\right)\) | \(e\left(\frac{7}{516}\right)\) | \(e\left(\frac{131}{172}\right)\) | \(e\left(\frac{813}{1204}\right)\) | \(e\left(\frac{274}{301}\right)\) | \(e\left(\frac{493}{602}\right)\) | \(e\left(\frac{691}{3612}\right)\) |
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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)
