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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([195,650,624,615]))
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gp:[g,chi] = znchar(Mod(10057, 65065))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65065.10057");
| 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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{65065}(47,\cdot)\)
\(\chi_{65065}(213,\cdot)\)
\(\chi_{65065}(278,\cdot)\)
\(\chi_{65065}(1347,\cdot)\)
\(\chi_{65065}(1412,\cdot)\)
\(\chi_{65065}(1578,\cdot)\)
\(\chi_{65065}(1643,\cdot)\)
\(\chi_{65065}(1802,\cdot)\)
\(\chi_{65065}(2033,\cdot)\)
\(\chi_{65065}(2777,\cdot)\)
\(\chi_{65065}(3008,\cdot)\)
\(\chi_{65065}(3232,\cdot)\)
\(\chi_{65065}(3463,\cdot)\)
\(\chi_{65065}(3622,\cdot)\)
\(\chi_{65065}(3853,\cdot)\)
\(\chi_{65065}(4987,\cdot)\)
\(\chi_{65065}(5052,\cdot)\)
\(\chi_{65065}(5218,\cdot)\)
\(\chi_{65065}(5283,\cdot)\)
\(\chi_{65065}(6417,\cdot)\)
\(\chi_{65065}(6583,\cdot)\)
\(\chi_{65065}(6648,\cdot)\)
\(\chi_{65065}(6807,\cdot)\)
\(\chi_{65065}(7038,\cdot)\)
\(\chi_{65065}(7782,\cdot)\)
\(\chi_{65065}(8237,\cdot)\)
\(\chi_{65065}(8468,\cdot)\)
\(\chi_{65065}(8627,\cdot)\)
\(\chi_{65065}(9992,\cdot)\)
\(\chi_{65065}(10057,\cdot)\)
...
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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 };
\((26027,46476,41406,6931)\) → \((i,e\left(\frac{5}{6}\right),e\left(\frac{4}{5}\right),e\left(\frac{41}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(16\) | \(17\) | \(18\) |
| \( \chi_{ 65065 }(10057, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{197}{390}\right)\) | \(e\left(\frac{587}{780}\right)\) | \(e\left(\frac{2}{195}\right)\) | \(e\left(\frac{67}{260}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{197}{390}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{4}{195}\right)\) | \(e\left(\frac{311}{780}\right)\) | \(e\left(\frac{2}{195}\right)\) |
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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)
