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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10033, base_ring=CyclotomicField(1638))
M = H._module
chi = DirichletCharacter(H, M([21,13]))
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gp:[g,chi] = znchar(Mod(3, 10033))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10033.3");
| Modulus: | \(10033\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10033\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1638\) |
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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);
|
\(\chi_{10033}(3,\cdot)\)
\(\chi_{10033}(29,\cdot)\)
\(\chi_{10033}(39,\cdot)\)
\(\chi_{10033}(53,\cdot)\)
\(\chi_{10033}(85,\cdot)\)
\(\chi_{10033}(86,\cdot)\)
\(\chi_{10033}(114,\cdot)\)
\(\chi_{10033}(118,\cdot)\)
\(\chi_{10033}(133,\cdot)\)
\(\chi_{10033}(192,\cdot)\)
\(\chi_{10033}(205,\cdot)\)
\(\chi_{10033}(218,\cdot)\)
\(\chi_{10033}(224,\cdot)\)
\(\chi_{10033}(233,\cdot)\)
\(\chi_{10033}(243,\cdot)\)
\(\chi_{10033}(297,\cdot)\)
\(\chi_{10033}(300,\cdot)\)
\(\chi_{10033}(307,\cdot)\)
\(\chi_{10033}(312,\cdot)\)
\(\chi_{10033}(319,\cdot)\)
\(\chi_{10033}(345,\cdot)\)
\(\chi_{10033}(346,\cdot)\)
\(\chi_{10033}(350,\cdot)\)
\(\chi_{10033}(351,\cdot)\)
\(\chi_{10033}(363,\cdot)\)
\(\chi_{10033}(364,\cdot)\)
\(\chi_{10033}(393,\cdot)\)
\(\chi_{10033}(424,\cdot)\)
\(\chi_{10033}(429,\cdot)\)
\(\chi_{10033}(434,\cdot)\)
...
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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 };
\((7113,2924)\) → \((e\left(\frac{1}{78}\right),e\left(\frac{1}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 10033 }(3, a) \) |
\(1\) | \(1\) | \(e\left(\frac{170}{273}\right)\) | \(e\left(\frac{17}{819}\right)\) | \(e\left(\frac{67}{273}\right)\) | \(e\left(\frac{265}{546}\right)\) | \(e\left(\frac{527}{819}\right)\) | \(e\left(\frac{485}{819}\right)\) | \(e\left(\frac{79}{91}\right)\) | \(e\left(\frac{34}{819}\right)\) | \(e\left(\frac{59}{546}\right)\) | \(e\left(\frac{337}{819}\right)\) |
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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)
