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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(34727, base_ring=CyclotomicField(440))
M = H._module
chi = DirichletCharacter(H, M([220,228,33]))
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gp:[g,chi] = znchar(Mod(6530, 34727))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("34727.6530");
| Modulus: | \(34727\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(34727\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(440\) |
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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);
|
\(\chi_{34727}(6,\cdot)\)
\(\chi_{34727}(216,\cdot)\)
\(\chi_{34727}(272,\cdot)\)
\(\chi_{34727}(321,\cdot)\)
\(\chi_{34727}(552,\cdot)\)
\(\chi_{34727}(755,\cdot)\)
\(\chi_{34727}(930,\cdot)\)
\(\chi_{34727}(1119,\cdot)\)
\(\chi_{34727}(1161,\cdot)\)
\(\chi_{34727}(1910,\cdot)\)
\(\chi_{34727}(2085,\cdot)\)
\(\chi_{34727}(2120,\cdot)\)
\(\chi_{34727}(2400,\cdot)\)
\(\chi_{34727}(2449,\cdot)\)
\(\chi_{34727}(2631,\cdot)\)
\(\chi_{34727}(2967,\cdot)\)
\(\chi_{34727}(3163,\cdot)\)
\(\chi_{34727}(3373,\cdot)\)
\(\chi_{34727}(3429,\cdot)\)
\(\chi_{34727}(3478,\cdot)\)
\(\chi_{34727}(3709,\cdot)\)
\(\chi_{34727}(4276,\cdot)\)
\(\chi_{34727}(4318,\cdot)\)
\(\chi_{34727}(5067,\cdot)\)
\(\chi_{34727}(5242,\cdot)\)
\(\chi_{34727}(5277,\cdot)\)
\(\chi_{34727}(5788,\cdot)\)
\(\chi_{34727}(6124,\cdot)\)
\(\chi_{34727}(6320,\cdot)\)
\(\chi_{34727}(6530,\cdot)\)
...
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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 };
\((29767,22387,22023)\) → \((-1,e\left(\frac{57}{110}\right),e\left(\frac{3}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 34727 }(6530, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{71}{440}\right)\) | \(e\left(\frac{71}{440}\right)\) |
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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)
