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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22860, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,42,63,50]))
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gp:[g,chi] = znchar(Mod(3019, 22860))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22860.3019");
| Modulus: | \(22860\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(22860\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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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\(\chi_{22860}(79,\cdot)\)
\(\chi_{22860}(679,\cdot)\)
\(\chi_{22860}(1939,\cdot)\)
\(\chi_{22860}(3019,\cdot)\)
\(\chi_{22860}(3499,\cdot)\)
\(\chi_{22860}(4279,\cdot)\)
\(\chi_{22860}(4399,\cdot)\)
\(\chi_{22860}(4759,\cdot)\)
\(\chi_{22860}(4939,\cdot)\)
\(\chi_{22860}(5659,\cdot)\)
\(\chi_{22860}(5839,\cdot)\)
\(\chi_{22860}(6259,\cdot)\)
\(\chi_{22860}(8359,\cdot)\)
\(\chi_{22860}(8539,\cdot)\)
\(\chi_{22860}(8959,\cdot)\)
\(\chi_{22860}(9079,\cdot)\)
\(\chi_{22860}(9259,\cdot)\)
\(\chi_{22860}(10699,\cdot)\)
\(\chi_{22860}(10939,\cdot)\)
\(\chi_{22860}(11479,\cdot)\)
\(\chi_{22860}(13099,\cdot)\)
\(\chi_{22860}(14179,\cdot)\)
\(\chi_{22860}(15439,\cdot)\)
\(\chi_{22860}(16519,\cdot)\)
\(\chi_{22860}(17059,\cdot)\)
\(\chi_{22860}(17539,\cdot)\)
\(\chi_{22860}(18499,\cdot)\)
\(\chi_{22860}(19219,\cdot)\)
\(\chi_{22860}(19339,\cdot)\)
\(\chi_{22860}(19579,\cdot)\)
...
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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 };
\((11431,12701,13717,4321)\) → \((-1,e\left(\frac{1}{3}\right),-1,e\left(\frac{25}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 22860 }(3019, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{26}{63}\right)\) |
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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)
