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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5049, base_ring=CyclotomicField(720))
M = H._module
chi = DirichletCharacter(H, M([280,216,225]))
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gp:[g,chi] = znchar(Mod(668, 5049))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5049.668");
| Modulus: | \(5049\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5049\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(720\) |
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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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\(\chi_{5049}(29,\cdot)\)
\(\chi_{5049}(41,\cdot)\)
\(\chi_{5049}(74,\cdot)\)
\(\chi_{5049}(95,\cdot)\)
\(\chi_{5049}(167,\cdot)\)
\(\chi_{5049}(173,\cdot)\)
\(\chi_{5049}(182,\cdot)\)
\(\chi_{5049}(194,\cdot)\)
\(\chi_{5049}(227,\cdot)\)
\(\chi_{5049}(248,\cdot)\)
\(\chi_{5049}(266,\cdot)\)
\(\chi_{5049}(299,\cdot)\)
\(\chi_{5049}(326,\cdot)\)
\(\chi_{5049}(347,\cdot)\)
\(\chi_{5049}(371,\cdot)\)
\(\chi_{5049}(380,\cdot)\)
\(\chi_{5049}(398,\cdot)\)
\(\chi_{5049}(437,\cdot)\)
\(\chi_{5049}(464,\cdot)\)
\(\chi_{5049}(470,\cdot)\)
\(\chi_{5049}(479,\cdot)\)
\(\chi_{5049}(524,\cdot)\)
\(\chi_{5049}(590,\cdot)\)
\(\chi_{5049}(623,\cdot)\)
\(\chi_{5049}(635,\cdot)\)
\(\chi_{5049}(668,\cdot)\)
\(\chi_{5049}(677,\cdot)\)
\(\chi_{5049}(734,\cdot)\)
\(\chi_{5049}(743,\cdot)\)
\(\chi_{5049}(776,\cdot)\)
...
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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 };
\((2432,3214,3862)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{3}{10}\right),e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 5049 }(668, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{360}\right)\) | \(e\left(\frac{23}{180}\right)\) | \(e\left(\frac{509}{720}\right)\) | \(e\left(\frac{547}{720}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{119}{180}\right)\) | \(e\left(\frac{593}{720}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{113}{120}\right)\) |
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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)
