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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65065, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([585,260,546,20]))
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gp:[g,chi] = znchar(Mod(4748, 65065))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65065.4748");
| Modulus: | \(65065\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(65065\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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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);
|
\(\chi_{65065}(107,\cdot)\)
\(\chi_{65065}(347,\cdot)\)
\(\chi_{65065}(893,\cdot)\)
\(\chi_{65065}(1108,\cdot)\)
\(\chi_{65065}(1348,\cdot)\)
\(\chi_{65065}(1927,\cdot)\)
\(\chi_{65065}(2382,\cdot)\)
\(\chi_{65065}(2713,\cdot)\)
\(\chi_{65065}(2928,\cdot)\)
\(\chi_{65065}(3077,\cdot)\)
\(\chi_{65065}(3383,\cdot)\)
\(\chi_{65065}(3747,\cdot)\)
\(\chi_{65065}(4748,\cdot)\)
\(\chi_{65065}(4897,\cdot)\)
\(\chi_{65065}(5112,\cdot)\)
\(\chi_{65065}(5352,\cdot)\)
\(\chi_{65065}(5898,\cdot)\)
\(\chi_{65065}(6113,\cdot)\)
\(\chi_{65065}(6353,\cdot)\)
\(\chi_{65065}(6717,\cdot)\)
\(\chi_{65065}(6932,\cdot)\)
\(\chi_{65065}(7387,\cdot)\)
\(\chi_{65065}(7718,\cdot)\)
\(\chi_{65065}(7933,\cdot)\)
\(\chi_{65065}(8082,\cdot)\)
\(\chi_{65065}(8388,\cdot)\)
\(\chi_{65065}(8752,\cdot)\)
\(\chi_{65065}(9083,\cdot)\)
\(\chi_{65065}(9753,\cdot)\)
\(\chi_{65065}(9902,\cdot)\)
...
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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 };
\((26027,46476,41406,6931)\) → \((-i,e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right),e\left(\frac{1}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(16\) | \(17\) | \(18\) |
| \( \chi_{ 65065 }(4748, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{260}\right)\) | \(e\left(\frac{283}{780}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{197}{390}\right)\) | \(e\left(\frac{111}{260}\right)\) | \(e\left(\frac{283}{390}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{33}{260}\right)\) | \(e\left(\frac{677}{780}\right)\) |
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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)
