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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1569, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([87,91]))
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gp:[g,chi] = znchar(Mod(611, 1569))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1569.611");
| Modulus: | \(1569\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1569\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(174\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{1569}(8,\cdot)\)
\(\chi_{1569}(14,\cdot)\)
\(\chi_{1569}(20,\cdot)\)
\(\chi_{1569}(26,\cdot)\)
\(\chi_{1569}(35,\cdot)\)
\(\chi_{1569}(38,\cdot)\)
\(\chi_{1569}(59,\cdot)\)
\(\chi_{1569}(62,\cdot)\)
\(\chi_{1569}(65,\cdot)\)
\(\chi_{1569}(101,\cdot)\)
\(\chi_{1569}(125,\cdot)\)
\(\chi_{1569}(188,\cdot)\)
\(\chi_{1569}(239,\cdot)\)
\(\chi_{1569}(269,\cdot)\)
\(\chi_{1569}(329,\cdot)\)
\(\chi_{1569}(344,\cdot)\)
\(\chi_{1569}(386,\cdot)\)
\(\chi_{1569}(470,\cdot)\)
\(\chi_{1569}(482,\cdot)\)
\(\chi_{1569}(506,\cdot)\)
\(\chi_{1569}(569,\cdot)\)
\(\chi_{1569}(602,\cdot)\)
\(\chi_{1569}(611,\cdot)\)
\(\chi_{1569}(668,\cdot)\)
\(\chi_{1569}(671,\cdot)\)
\(\chi_{1569}(677,\cdot)\)
\(\chi_{1569}(743,\cdot)\)
\(\chi_{1569}(782,\cdot)\)
\(\chi_{1569}(809,\cdot)\)
\(\chi_{1569}(830,\cdot)\)
...
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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 };
\((524,1048)\) → \((-1,e\left(\frac{91}{174}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1569 }(611, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{87}\right)\) | \(e\left(\frac{4}{87}\right)\) | \(e\left(\frac{11}{87}\right)\) | \(e\left(\frac{82}{87}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{13}{87}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{55}{87}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{8}{87}\right)\) |
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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)
