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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29575, base_ring=CyclotomicField(390))
M = H._module
chi = DirichletCharacter(H, M([39,65,295]))
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gp:[g,chi] = znchar(Mod(5379, 29575))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29575.5379");
| Modulus: | \(29575\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(29575\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(390\) |
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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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\(\chi_{29575}(264,\cdot)\)
\(\chi_{29575}(719,\cdot)\)
\(\chi_{29575}(829,\cdot)\)
\(\chi_{29575}(1284,\cdot)\)
\(\chi_{29575}(1629,\cdot)\)
\(\chi_{29575}(1739,\cdot)\)
\(\chi_{29575}(2084,\cdot)\)
\(\chi_{29575}(2194,\cdot)\)
\(\chi_{29575}(2539,\cdot)\)
\(\chi_{29575}(2994,\cdot)\)
\(\chi_{29575}(3104,\cdot)\)
\(\chi_{29575}(3559,\cdot)\)
\(\chi_{29575}(3904,\cdot)\)
\(\chi_{29575}(4014,\cdot)\)
\(\chi_{29575}(4359,\cdot)\)
\(\chi_{29575}(4469,\cdot)\)
\(\chi_{29575}(4814,\cdot)\)
\(\chi_{29575}(5269,\cdot)\)
\(\chi_{29575}(5379,\cdot)\)
\(\chi_{29575}(5834,\cdot)\)
\(\chi_{29575}(6179,\cdot)\)
\(\chi_{29575}(6289,\cdot)\)
\(\chi_{29575}(6634,\cdot)\)
\(\chi_{29575}(6744,\cdot)\)
\(\chi_{29575}(7089,\cdot)\)
\(\chi_{29575}(7544,\cdot)\)
\(\chi_{29575}(7654,\cdot)\)
\(\chi_{29575}(8109,\cdot)\)
\(\chi_{29575}(8454,\cdot)\)
\(\chi_{29575}(8564,\cdot)\)
...
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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 };
\((26027,16901,24676)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{1}{6}\right),e\left(\frac{59}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(16\) | \(17\) |
| \( \chi_{ 29575 }(5379, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{195}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{74}{195}\right)\) | \(e\left(\frac{166}{195}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{8}{195}\right)\) | \(e\left(\frac{148}{195}\right)\) | \(e\left(\frac{176}{195}\right)\) |
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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)
