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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(59245, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([51,7,34]))
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gp:[g,chi] = znchar(Mod(54693, 59245))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("59245.54693");
| Modulus: | \(59245\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(59245\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(68\) |
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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);
|
\(\chi_{59245}(2172,\cdot)\)
\(\chi_{59245}(2418,\cdot)\)
\(\chi_{59245}(5657,\cdot)\)
\(\chi_{59245}(5903,\cdot)\)
\(\chi_{59245}(9142,\cdot)\)
\(\chi_{59245}(9388,\cdot)\)
\(\chi_{59245}(12627,\cdot)\)
\(\chi_{59245}(12873,\cdot)\)
\(\chi_{59245}(16112,\cdot)\)
\(\chi_{59245}(16358,\cdot)\)
\(\chi_{59245}(19597,\cdot)\)
\(\chi_{59245}(19843,\cdot)\)
\(\chi_{59245}(23328,\cdot)\)
\(\chi_{59245}(26567,\cdot)\)
\(\chi_{59245}(26813,\cdot)\)
\(\chi_{59245}(30052,\cdot)\)
\(\chi_{59245}(30298,\cdot)\)
\(\chi_{59245}(33537,\cdot)\)
\(\chi_{59245}(33783,\cdot)\)
\(\chi_{59245}(37022,\cdot)\)
\(\chi_{59245}(37268,\cdot)\)
\(\chi_{59245}(40507,\cdot)\)
\(\chi_{59245}(40753,\cdot)\)
\(\chi_{59245}(43992,\cdot)\)
\(\chi_{59245}(44238,\cdot)\)
\(\chi_{59245}(47477,\cdot)\)
\(\chi_{59245}(50962,\cdot)\)
\(\chi_{59245}(51208,\cdot)\)
\(\chi_{59245}(54447,\cdot)\)
\(\chi_{59245}(54693,\cdot)\)
...
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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 };
\((47397,35261,30346)\) → \((-i,e\left(\frac{7}{68}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 59245 }(54693, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{63}{68}\right)\) |
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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)
