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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91035, base_ring=CyclotomicField(816))
M = H._module
chi = DirichletCharacter(H, M([680,612,544,567]))
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gp:[g,chi] = znchar(Mod(9923, 91035))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91035.9923");
| Modulus: | \(91035\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(91035\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(816\) |
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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);
|
\(\chi_{91035}(317,\cdot)\)
\(\chi_{91035}(347,\cdot)\)
\(\chi_{91035}(473,\cdot)\)
\(\chi_{91035}(1703,\cdot)\)
\(\chi_{91035}(2018,\cdot)\)
\(\chi_{91035}(2522,\cdot)\)
\(\chi_{91035}(2867,\cdot)\)
\(\chi_{91035}(2963,\cdot)\)
\(\chi_{91035}(3152,\cdot)\)
\(\chi_{91035}(3278,\cdot)\)
\(\chi_{91035}(4253,\cdot)\)
\(\chi_{91035}(4568,\cdot)\)
\(\chi_{91035}(5042,\cdot)\)
\(\chi_{91035}(5072,\cdot)\)
\(\chi_{91035}(5513,\cdot)\)
\(\chi_{91035}(5672,\cdot)\)
\(\chi_{91035}(5702,\cdot)\)
\(\chi_{91035}(5828,\cdot)\)
\(\chi_{91035}(7058,\cdot)\)
\(\chi_{91035}(7373,\cdot)\)
\(\chi_{91035}(7592,\cdot)\)
\(\chi_{91035}(7877,\cdot)\)
\(\chi_{91035}(8222,\cdot)\)
\(\chi_{91035}(8318,\cdot)\)
\(\chi_{91035}(8507,\cdot)\)
\(\chi_{91035}(8633,\cdot)\)
\(\chi_{91035}(9608,\cdot)\)
\(\chi_{91035}(9923,\cdot)\)
\(\chi_{91035}(10397,\cdot)\)
\(\chi_{91035}(10427,\cdot)\)
...
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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 };
\((80921,54622,65026,28036)\) → \((e\left(\frac{5}{6}\right),-i,e\left(\frac{2}{3}\right),e\left(\frac{189}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 91035 }(9923, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{383}{408}\right)\) | \(e\left(\frac{179}{204}\right)\) | \(e\left(\frac{111}{136}\right)\) | \(e\left(\frac{131}{272}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{229}{408}\right)\) | \(e\left(\frac{343}{816}\right)\) | \(e\left(\frac{191}{272}\right)\) | \(e\left(\frac{19}{408}\right)\) |
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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)
