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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2101, base_ring=CyclotomicField(190))
M = H._module
chi = DirichletCharacter(H, M([152,54]))
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gp:[g,chi] = znchar(Mod(1059, 2101))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2101.1059");
| Modulus: | \(2101\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2101\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(95\) |
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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);
|
\(\chi_{2101}(15,\cdot)\)
\(\chi_{2101}(20,\cdot)\)
\(\chi_{2101}(26,\cdot)\)
\(\chi_{2101}(158,\cdot)\)
\(\chi_{2101}(170,\cdot)\)
\(\chi_{2101}(201,\cdot)\)
\(\chi_{2101}(218,\cdot)\)
\(\chi_{2101}(225,\cdot)\)
\(\chi_{2101}(236,\cdot)\)
\(\chi_{2101}(268,\cdot)\)
\(\chi_{2101}(269,\cdot)\)
\(\chi_{2101}(324,\cdot)\)
\(\chi_{2101}(390,\cdot)\)
\(\chi_{2101}(400,\cdot)\)
\(\chi_{2101}(432,\cdot)\)
\(\chi_{2101}(449,\cdot)\)
\(\chi_{2101}(454,\cdot)\)
\(\chi_{2101}(482,\cdot)\)
\(\chi_{2101}(520,\cdot)\)
\(\chi_{2101}(526,\cdot)\)
\(\chi_{2101}(554,\cdot)\)
\(\chi_{2101}(575,\cdot)\)
\(\chi_{2101}(576,\cdot)\)
\(\chi_{2101}(621,\cdot)\)
\(\chi_{2101}(653,\cdot)\)
\(\chi_{2101}(658,\cdot)\)
\(\chi_{2101}(665,\cdot)\)
\(\chi_{2101}(676,\cdot)\)
\(\chi_{2101}(691,\cdot)\)
\(\chi_{2101}(735,\cdot)\)
...
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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 };
\((574,210)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{27}{95}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2101 }(1059, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{95}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{58}{95}\right)\) | \(e\left(\frac{39}{95}\right)\) | \(e\left(\frac{64}{95}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{87}{95}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{68}{95}\right)\) | \(e\left(\frac{93}{95}\right)\) |
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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)
