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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(36703, base_ring=CyclotomicField(306))
M = H._module
chi = DirichletCharacter(H, M([225,17]))
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gp:[g,chi] = znchar(Mod(3076, 36703))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("36703.3076");
| Modulus: | \(36703\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(36703\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(306\) |
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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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\(\chi_{36703}(186,\cdot)\)
\(\chi_{36703}(917,\cdot)\)
\(\chi_{36703}(1121,\cdot)\)
\(\chi_{36703}(1614,\cdot)\)
\(\chi_{36703}(2056,\cdot)\)
\(\chi_{36703}(2107,\cdot)\)
\(\chi_{36703}(2345,\cdot)\)
\(\chi_{36703}(3076,\cdot)\)
\(\chi_{36703}(3280,\cdot)\)
\(\chi_{36703}(3773,\cdot)\)
\(\chi_{36703}(4215,\cdot)\)
\(\chi_{36703}(4266,\cdot)\)
\(\chi_{36703}(4504,\cdot)\)
\(\chi_{36703}(5235,\cdot)\)
\(\chi_{36703}(5439,\cdot)\)
\(\chi_{36703}(5932,\cdot)\)
\(\chi_{36703}(6374,\cdot)\)
\(\chi_{36703}(6425,\cdot)\)
\(\chi_{36703}(6663,\cdot)\)
\(\chi_{36703}(7394,\cdot)\)
\(\chi_{36703}(7598,\cdot)\)
\(\chi_{36703}(8533,\cdot)\)
\(\chi_{36703}(8584,\cdot)\)
\(\chi_{36703}(8822,\cdot)\)
\(\chi_{36703}(9553,\cdot)\)
\(\chi_{36703}(9757,\cdot)\)
\(\chi_{36703}(10250,\cdot)\)
\(\chi_{36703}(10743,\cdot)\)
\(\chi_{36703}(11712,\cdot)\)
\(\chi_{36703}(11916,\cdot)\)
...
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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 };
\((16765,19942)\) → \((e\left(\frac{25}{34}\right),e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 36703 }(3076, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{121}{153}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{76}{153}\right)\) | \(e\left(\frac{55}{153}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{89}{153}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{211}{306}\right)\) |
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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)
