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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38808, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,105,175,150,147]))
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gp:[g,chi] = znchar(Mod(1877, 38808))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38808.1877");
| Modulus: | \(38808\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(38808\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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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\(\chi_{38808}(29,\cdot)\)
\(\chi_{38808}(365,\cdot)\)
\(\chi_{38808}(1877,\cdot)\)
\(\chi_{38808}(3053,\cdot)\)
\(\chi_{38808}(4061,\cdot)\)
\(\chi_{38808}(4397,\cdot)\)
\(\chi_{38808}(5573,\cdot)\)
\(\chi_{38808}(5909,\cdot)\)
\(\chi_{38808}(7421,\cdot)\)
\(\chi_{38808}(8093,\cdot)\)
\(\chi_{38808}(8597,\cdot)\)
\(\chi_{38808}(9941,\cdot)\)
\(\chi_{38808}(10445,\cdot)\)
\(\chi_{38808}(11117,\cdot)\)
\(\chi_{38808}(11453,\cdot)\)
\(\chi_{38808}(12965,\cdot)\)
\(\chi_{38808}(13637,\cdot)\)
\(\chi_{38808}(14141,\cdot)\)
\(\chi_{38808}(15149,\cdot)\)
\(\chi_{38808}(15989,\cdot)\)
\(\chi_{38808}(16997,\cdot)\)
\(\chi_{38808}(18509,\cdot)\)
\(\chi_{38808}(19181,\cdot)\)
\(\chi_{38808}(19685,\cdot)\)
\(\chi_{38808}(20693,\cdot)\)
\(\chi_{38808}(21029,\cdot)\)
\(\chi_{38808}(21533,\cdot)\)
\(\chi_{38808}(22205,\cdot)\)
\(\chi_{38808}(24053,\cdot)\)
\(\chi_{38808}(24725,\cdot)\)
...
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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 };
\((9703,19405,4313,29305,24697)\) → \((1,-1,e\left(\frac{5}{6}\right),e\left(\frac{5}{7}\right),e\left(\frac{7}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 38808 }(1877, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{103}{105}\right)\) |
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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)
