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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2585, base_ring=CyclotomicField(230))
M = H._module
chi = DirichletCharacter(H, M([115,46,125]))
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gp:[g,chi] = znchar(Mod(774, 2585))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2585.774");
| Modulus: | \(2585\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2585\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(230\) |
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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: | odd |
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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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\(\chi_{2585}(69,\cdot)\)
\(\chi_{2585}(104,\cdot)\)
\(\chi_{2585}(114,\cdot)\)
\(\chi_{2585}(124,\cdot)\)
\(\chi_{2585}(174,\cdot)\)
\(\chi_{2585}(179,\cdot)\)
\(\chi_{2585}(214,\cdot)\)
\(\chi_{2585}(229,\cdot)\)
\(\chi_{2585}(279,\cdot)\)
\(\chi_{2585}(334,\cdot)\)
\(\chi_{2585}(339,\cdot)\)
\(\chi_{2585}(344,\cdot)\)
\(\chi_{2585}(389,\cdot)\)
\(\chi_{2585}(399,\cdot)\)
\(\chi_{2585}(434,\cdot)\)
\(\chi_{2585}(449,\cdot)\)
\(\chi_{2585}(454,\cdot)\)
\(\chi_{2585}(489,\cdot)\)
\(\chi_{2585}(499,\cdot)\)
\(\chi_{2585}(509,\cdot)\)
\(\chi_{2585}(599,\cdot)\)
\(\chi_{2585}(609,\cdot)\)
\(\chi_{2585}(654,\cdot)\)
\(\chi_{2585}(669,\cdot)\)
\(\chi_{2585}(724,\cdot)\)
\(\chi_{2585}(774,\cdot)\)
\(\chi_{2585}(819,\cdot)\)
\(\chi_{2585}(829,\cdot)\)
\(\chi_{2585}(834,\cdot)\)
\(\chi_{2585}(839,\cdot)\)
...
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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 };
\((1552,706,1321)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{25}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 2585 }(774, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{111}{230}\right)\) | \(e\left(\frac{223}{230}\right)\) | \(e\left(\frac{111}{115}\right)\) | \(e\left(\frac{52}{115}\right)\) | \(e\left(\frac{67}{230}\right)\) | \(e\left(\frac{103}{230}\right)\) | \(e\left(\frac{108}{115}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{78}{115}\right)\) | \(e\left(\frac{89}{115}\right)\) |
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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)
