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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6363, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([125,25,63]))
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gp:[g,chi] = znchar(Mod(1760, 6363))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6363.1760");
| Modulus: | \(6363\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6363\) |
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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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\(\chi_{6363}(47,\cdot)\)
\(\chi_{6363}(110,\cdot)\)
\(\chi_{6363}(122,\cdot)\)
\(\chi_{6363}(425,\cdot)\)
\(\chi_{6363}(437,\cdot)\)
\(\chi_{6363}(500,\cdot)\)
\(\chi_{6363}(626,\cdot)\)
\(\chi_{6363}(740,\cdot)\)
\(\chi_{6363}(752,\cdot)\)
\(\chi_{6363}(803,\cdot)\)
\(\chi_{6363}(878,\cdot)\)
\(\chi_{6363}(929,\cdot)\)
\(\chi_{6363}(1055,\cdot)\)
\(\chi_{6363}(1181,\cdot)\)
\(\chi_{6363}(1193,\cdot)\)
\(\chi_{6363}(1496,\cdot)\)
\(\chi_{6363}(1760,\cdot)\)
\(\chi_{6363}(1949,\cdot)\)
\(\chi_{6363}(2063,\cdot)\)
\(\chi_{6363}(2252,\cdot)\)
\(\chi_{6363}(2327,\cdot)\)
\(\chi_{6363}(2630,\cdot)\)
\(\chi_{6363}(3713,\cdot)\)
\(\chi_{6363}(3902,\cdot)\)
\(\chi_{6363}(4016,\cdot)\)
\(\chi_{6363}(4154,\cdot)\)
\(\chi_{6363}(4205,\cdot)\)
\(\chi_{6363}(4217,\cdot)\)
\(\chi_{6363}(4457,\cdot)\)
\(\chi_{6363}(4520,\cdot)\)
...
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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 };
\((1415,4546,5860)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{6}\right),e\left(\frac{21}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 6363 }(1760, a) \) |
\(1\) | \(1\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{150}\right)\) |
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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)
