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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22815, base_ring=CyclotomicField(468))
M = H._module
chi = DirichletCharacter(H, M([338,117,165]))
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gp:[g,chi] = znchar(Mod(3872, 22815))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22815.3872");
| Modulus: | \(22815\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(22815\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(468\) |
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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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\(\chi_{22815}(353,\cdot)\)
\(\chi_{22815}(362,\cdot)\)
\(\chi_{22815}(383,\cdot)\)
\(\chi_{22815}(527,\cdot)\)
\(\chi_{22815}(938,\cdot)\)
\(\chi_{22815}(947,\cdot)\)
\(\chi_{22815}(968,\cdot)\)
\(\chi_{22815}(1112,\cdot)\)
\(\chi_{22815}(1523,\cdot)\)
\(\chi_{22815}(1532,\cdot)\)
\(\chi_{22815}(1553,\cdot)\)
\(\chi_{22815}(1697,\cdot)\)
\(\chi_{22815}(2138,\cdot)\)
\(\chi_{22815}(2282,\cdot)\)
\(\chi_{22815}(2693,\cdot)\)
\(\chi_{22815}(2702,\cdot)\)
\(\chi_{22815}(2867,\cdot)\)
\(\chi_{22815}(3278,\cdot)\)
\(\chi_{22815}(3287,\cdot)\)
\(\chi_{22815}(3308,\cdot)\)
\(\chi_{22815}(3452,\cdot)\)
\(\chi_{22815}(3863,\cdot)\)
\(\chi_{22815}(3872,\cdot)\)
\(\chi_{22815}(3893,\cdot)\)
\(\chi_{22815}(4448,\cdot)\)
\(\chi_{22815}(4457,\cdot)\)
\(\chi_{22815}(4478,\cdot)\)
\(\chi_{22815}(4622,\cdot)\)
\(\chi_{22815}(5033,\cdot)\)
\(\chi_{22815}(5042,\cdot)\)
...
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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 };
\((14366,9127,22141)\) → \((e\left(\frac{13}{18}\right),i,e\left(\frac{55}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 22815 }(3872, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{38}{117}\right)\) | \(e\left(\frac{76}{117}\right)\) | \(e\left(\frac{62}{117}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{329}{468}\right)\) | \(e\left(\frac{100}{117}\right)\) | \(e\left(\frac{35}{117}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{36}\right)\) |
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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)
