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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3397, base_ring=CyclotomicField(546))
M = H._module
chi = DirichletCharacter(H, M([377,140]))
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gp:[g,chi] = znchar(Mod(190, 3397))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3397.190");
| Modulus: | \(3397\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3397\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(546\) |
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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);
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\(\chi_{3397}(20,\cdot)\)
\(\chi_{3397}(26,\cdot)\)
\(\chi_{3397}(72,\cdot)\)
\(\chi_{3397}(73,\cdot)\)
\(\chi_{3397}(76,\cdot)\)
\(\chi_{3397}(163,\cdot)\)
\(\chi_{3397}(190,\cdot)\)
\(\chi_{3397}(198,\cdot)\)
\(\chi_{3397}(234,\cdot)\)
\(\chi_{3397}(241,\cdot)\)
\(\chi_{3397}(287,\cdot)\)
\(\chi_{3397}(313,\cdot)\)
\(\chi_{3397}(327,\cdot)\)
\(\chi_{3397}(335,\cdot)\)
\(\chi_{3397}(392,\cdot)\)
\(\chi_{3397}(420,\cdot)\)
\(\chi_{3397}(421,\cdot)\)
\(\chi_{3397}(476,\cdot)\)
\(\chi_{3397}(519,\cdot)\)
\(\chi_{3397}(546,\cdot)\)
\(\chi_{3397}(578,\cdot)\)
\(\chi_{3397}(585,\cdot)\)
\(\chi_{3397}(593,\cdot)\)
\(\chi_{3397}(636,\cdot)\)
\(\chi_{3397}(648,\cdot)\)
\(\chi_{3397}(657,\cdot)\)
\(\chi_{3397}(663,\cdot)\)
\(\chi_{3397}(708,\cdot)\)
\(\chi_{3397}(716,\cdot)\)
\(\chi_{3397}(722,\cdot)\)
...
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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 };
\((949,2452)\) → \((e\left(\frac{29}{42}\right),e\left(\frac{10}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 3397 }(190, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{365}{546}\right)\) | \(e\left(\frac{517}{546}\right)\) | \(e\left(\frac{92}{273}\right)\) | \(e\left(\frac{29}{182}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{1}{182}\right)\) | \(e\left(\frac{244}{273}\right)\) | \(e\left(\frac{226}{273}\right)\) | \(e\left(\frac{41}{273}\right)\) |
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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)
