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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74005, base_ring=CyclotomicField(6840))
M = H._module
chi = DirichletCharacter(H, M([3420,2540,1197]))
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gp:[g,chi] = znchar(Mod(439, 74005))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("74005.439");
| Modulus: | \(74005\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(74005\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(6840\) |
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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);
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\(\chi_{74005}(29,\cdot)\)
\(\chi_{74005}(34,\cdot)\)
\(\chi_{74005}(89,\cdot)\)
\(\chi_{74005}(129,\cdot)\)
\(\chi_{74005}(224,\cdot)\)
\(\chi_{74005}(439,\cdot)\)
\(\chi_{74005}(504,\cdot)\)
\(\chi_{74005}(509,\cdot)\)
\(\chi_{74005}(604,\cdot)\)
\(\chi_{74005}(649,\cdot)\)
\(\chi_{74005}(744,\cdot)\)
\(\chi_{74005}(794,\cdot)\)
\(\chi_{74005}(839,\cdot)\)
\(\chi_{74005}(889,\cdot)\)
\(\chi_{74005}(914,\cdot)\)
\(\chi_{74005}(1059,\cdot)\)
\(\chi_{74005}(1079,\cdot)\)
\(\chi_{74005}(1124,\cdot)\)
\(\chi_{74005}(1154,\cdot)\)
\(\chi_{74005}(1174,\cdot)\)
\(\chi_{74005}(1219,\cdot)\)
\(\chi_{74005}(1249,\cdot)\)
\(\chi_{74005}(1264,\cdot)\)
\(\chi_{74005}(1324,\cdot)\)
\(\chi_{74005}(1359,\cdot)\)
\(\chi_{74005}(1364,\cdot)\)
\(\chi_{74005}(1409,\cdot)\)
\(\chi_{74005}(1454,\cdot)\)
\(\chi_{74005}(1459,\cdot)\)
\(\chi_{74005}(1504,\cdot)\)
...
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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 };
\((14802,57401,9026)\) → \((-1,e\left(\frac{127}{342}\right),e\left(\frac{7}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 74005 }(439, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1441}{3420}\right)\) | \(e\left(\frac{1015}{1368}\right)\) | \(e\left(\frac{1441}{1710}\right)\) | \(e\left(\frac{1117}{6840}\right)\) | \(e\left(\frac{61}{2280}\right)\) | \(e\left(\frac{301}{1140}\right)\) | \(e\left(\frac{331}{684}\right)\) | \(e\left(\frac{917}{2280}\right)\) | \(e\left(\frac{1333}{2280}\right)\) | \(e\left(\frac{667}{6840}\right)\) |
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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)
