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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1313, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([100,51]))
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gp:[g,chi] = znchar(Mod(373, 1313))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1313.373");
| Modulus: | \(1313\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1313\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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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\(\chi_{1313}(9,\cdot)\)
\(\chi_{1313}(22,\cdot)\)
\(\chi_{1313}(146,\cdot)\)
\(\chi_{1313}(165,\cdot)\)
\(\chi_{1313}(178,\cdot)\)
\(\chi_{1313}(211,\cdot)\)
\(\chi_{1313}(224,\cdot)\)
\(\chi_{1313}(367,\cdot)\)
\(\chi_{1313}(373,\cdot)\)
\(\chi_{1313}(380,\cdot)\)
\(\chi_{1313}(399,\cdot)\)
\(\chi_{1313}(425,\cdot)\)
\(\chi_{1313}(451,\cdot)\)
\(\chi_{1313}(575,\cdot)\)
\(\chi_{1313}(581,\cdot)\)
\(\chi_{1313}(601,\cdot)\)
\(\chi_{1313}(627,\cdot)\)
\(\chi_{1313}(653,\cdot)\)
\(\chi_{1313}(711,\cdot)\)
\(\chi_{1313}(737,\cdot)\)
\(\chi_{1313}(750,\cdot)\)
\(\chi_{1313}(783,\cdot)\)
\(\chi_{1313}(789,\cdot)\)
\(\chi_{1313}(828,\cdot)\)
\(\chi_{1313}(841,\cdot)\)
\(\chi_{1313}(893,\cdot)\)
\(\chi_{1313}(913,\cdot)\)
\(\chi_{1313}(932,\cdot)\)
\(\chi_{1313}(939,\cdot)\)
\(\chi_{1313}(952,\cdot)\)
...
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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 };
\((405,911)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{17}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1313 }(373, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{59}{150}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{150}\right)\) |
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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)
