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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(64675, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([297,110,10]))
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gp:[g,chi] = znchar(Mod(17644, 64675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64675.17644");
| Modulus: | \(64675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(64675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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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\(\chi_{64675}(204,\cdot)\)
\(\chi_{64675}(1504,\cdot)\)
\(\chi_{64675}(1764,\cdot)\)
\(\chi_{64675}(1914,\cdot)\)
\(\chi_{64675}(3129,\cdot)\)
\(\chi_{64675}(4319,\cdot)\)
\(\chi_{64675}(4494,\cdot)\)
\(\chi_{64675}(4709,\cdot)\)
\(\chi_{64675}(7679,\cdot)\)
\(\chi_{64675}(8609,\cdot)\)
\(\chi_{64675}(8739,\cdot)\)
\(\chi_{64675}(8784,\cdot)\)
\(\chi_{64675}(9194,\cdot)\)
\(\chi_{64675}(9759,\cdot)\)
\(\chi_{64675}(11729,\cdot)\)
\(\chi_{64675}(11839,\cdot)\)
\(\chi_{64675}(12229,\cdot)\)
\(\chi_{64675}(13139,\cdot)\)
\(\chi_{64675}(13159,\cdot)\)
\(\chi_{64675}(14439,\cdot)\)
\(\chi_{64675}(16064,\cdot)\)
\(\chi_{64675}(17254,\cdot)\)
\(\chi_{64675}(17429,\cdot)\)
\(\chi_{64675}(17644,\cdot)\)
\(\chi_{64675}(19659,\cdot)\)
\(\chi_{64675}(20614,\cdot)\)
\(\chi_{64675}(20959,\cdot)\)
\(\chi_{64675}(21544,\cdot)\)
\(\chi_{64675}(21719,\cdot)\)
\(\chi_{64675}(22129,\cdot)\)
...
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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 };
\((59502,14926,45176)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{1}{3}\right),e\left(\frac{1}{33}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 64675 }(17644, a) \) |
\(1\) | \(1\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{151}{165}\right)\) |
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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)
