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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48884, base_ring=CyclotomicField(550))
M = H._module
chi = DirichletCharacter(H, M([275,40,209]))
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gp:[g,chi] = znchar(Mod(3039, 48884))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48884.3039");
| Modulus: | \(48884\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(48884\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(550\) |
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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);
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\(\chi_{48884}(47,\cdot)\)
\(\chi_{48884}(223,\cdot)\)
\(\chi_{48884}(235,\cdot)\)
\(\chi_{48884}(279,\cdot)\)
\(\chi_{48884}(427,\cdot)\)
\(\chi_{48884}(851,\cdot)\)
\(\chi_{48884}(1059,\cdot)\)
\(\chi_{48884}(1115,\cdot)\)
\(\chi_{48884}(1131,\cdot)\)
\(\chi_{48884}(1175,\cdot)\)
\(\chi_{48884}(1395,\cdot)\)
\(\chi_{48884}(1611,\cdot)\)
\(\chi_{48884}(1787,\cdot)\)
\(\chi_{48884}(1863,\cdot)\)
\(\chi_{48884}(2711,\cdot)\)
\(\chi_{48884}(3039,\cdot)\)
\(\chi_{48884}(3611,\cdot)\)
\(\chi_{48884}(3767,\cdot)\)
\(\chi_{48884}(4163,\cdot)\)
\(\chi_{48884}(4255,\cdot)\)
\(\chi_{48884}(4491,\cdot)\)
\(\chi_{48884}(4667,\cdot)\)
\(\chi_{48884}(4723,\cdot)\)
\(\chi_{48884}(4871,\cdot)\)
\(\chi_{48884}(5295,\cdot)\)
\(\chi_{48884}(5503,\cdot)\)
\(\chi_{48884}(5559,\cdot)\)
\(\chi_{48884}(5619,\cdot)\)
\(\chi_{48884}(5839,\cdot)\)
\(\chi_{48884}(6055,\cdot)\)
...
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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 };
\((24443,25049,11617)\) → \((-1,e\left(\frac{4}{55}\right),e\left(\frac{19}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 48884 }(3039, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{138}{275}\right)\) | \(e\left(\frac{118}{275}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{117}{275}\right)\) | \(e\left(\frac{171}{275}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{9}{550}\right)\) | \(e\left(\frac{151}{275}\right)\) | \(e\left(\frac{149}{550}\right)\) |
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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)
