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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5957, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([22,120,99]))
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gp:[g,chi] = znchar(Mod(1116, 5957))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5957.1116");
| Modulus: | \(5957\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5957\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(132\) |
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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);
|
\(\chi_{5957}(31,\cdot)\)
\(\chi_{5957}(117,\cdot)\)
\(\chi_{5957}(376,\cdot)\)
\(\chi_{5957}(808,\cdot)\)
\(\chi_{5957}(857,\cdot)\)
\(\chi_{5957}(1067,\cdot)\)
\(\chi_{5957}(1116,\cdot)\)
\(\chi_{5957}(1153,\cdot)\)
\(\chi_{5957}(1363,\cdot)\)
\(\chi_{5957}(1375,\cdot)\)
\(\chi_{5957}(1412,\cdot)\)
\(\chi_{5957}(1622,\cdot)\)
\(\chi_{5957}(1844,\cdot)\)
\(\chi_{5957}(1881,\cdot)\)
\(\chi_{5957}(2152,\cdot)\)
\(\chi_{5957}(2189,\cdot)\)
\(\chi_{5957}(2362,\cdot)\)
\(\chi_{5957}(2658,\cdot)\)
\(\chi_{5957}(2670,\cdot)\)
\(\chi_{5957}(2707,\cdot)\)
\(\chi_{5957}(2929,\cdot)\)
\(\chi_{5957}(3176,\cdot)\)
\(\chi_{5957}(3435,\cdot)\)
\(\chi_{5957}(3706,\cdot)\)
\(\chi_{5957}(3916,\cdot)\)
\(\chi_{5957}(3965,\cdot)\)
\(\chi_{5957}(4175,\cdot)\)
\(\chi_{5957}(4212,\cdot)\)
\(\chi_{5957}(4261,\cdot)\)
\(\chi_{5957}(4434,\cdot)\)
...
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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 };
\((3405,4145,3221)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{10}{11}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 5957 }(1116, a) \) |
\(1\) | \(1\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) |
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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)
