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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5925, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([65,91,60]))
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gp:[g,chi] = znchar(Mod(1934, 5925))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5925.1934");
| Modulus: | \(5925\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5925\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(130\) |
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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);
|
\(\chi_{5925}(89,\cdot)\)
\(\chi_{5925}(179,\cdot)\)
\(\chi_{5925}(539,\cdot)\)
\(\chi_{5925}(719,\cdot)\)
\(\chi_{5925}(854,\cdot)\)
\(\chi_{5925}(1079,\cdot)\)
\(\chi_{5925}(1094,\cdot)\)
\(\chi_{5925}(1364,\cdot)\)
\(\chi_{5925}(1484,\cdot)\)
\(\chi_{5925}(1784,\cdot)\)
\(\chi_{5925}(1904,\cdot)\)
\(\chi_{5925}(1934,\cdot)\)
\(\chi_{5925}(2039,\cdot)\)
\(\chi_{5925}(2234,\cdot)\)
\(\chi_{5925}(2264,\cdot)\)
\(\chi_{5925}(2279,\cdot)\)
\(\chi_{5925}(2309,\cdot)\)
\(\chi_{5925}(2459,\cdot)\)
\(\chi_{5925}(2669,\cdot)\)
\(\chi_{5925}(2909,\cdot)\)
\(\chi_{5925}(2969,\cdot)\)
\(\chi_{5925}(3089,\cdot)\)
\(\chi_{5925}(3119,\cdot)\)
\(\chi_{5925}(3419,\cdot)\)
\(\chi_{5925}(3464,\cdot)\)
\(\chi_{5925}(3494,\cdot)\)
\(\chi_{5925}(3644,\cdot)\)
\(\chi_{5925}(3734,\cdot)\)
\(\chi_{5925}(3854,\cdot)\)
\(\chi_{5925}(4094,\cdot)\)
...
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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 };
\((1976,5452,2926)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{6}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5925 }(1934, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) |
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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)
