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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40051, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([9,146]))
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gp:[g,chi] = znchar(Mod(11624, 40051))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40051.11624");
| Modulus: | \(40051\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(40051\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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);
|
\(\chi_{40051}(194,\cdot)\)
\(\chi_{40051}(271,\cdot)\)
\(\chi_{40051}(348,\cdot)\)
\(\chi_{40051}(514,\cdot)\)
\(\chi_{40051}(651,\cdot)\)
\(\chi_{40051}(744,\cdot)\)
\(\chi_{40051}(1014,\cdot)\)
\(\chi_{40051}(4864,\cdot)\)
\(\chi_{40051}(4879,\cdot)\)
\(\chi_{40051}(4930,\cdot)\)
\(\chi_{40051}(4974,\cdot)\)
\(\chi_{40051}(5198,\cdot)\)
\(\chi_{40051}(6107,\cdot)\)
\(\chi_{40051}(6309,\cdot)\)
\(\chi_{40051}(6729,\cdot)\)
\(\chi_{40051}(6804,\cdot)\)
\(\chi_{40051}(6892,\cdot)\)
\(\chi_{40051}(7563,\cdot)\)
\(\chi_{40051}(8234,\cdot)\)
\(\chi_{40051}(9345,\cdot)\)
\(\chi_{40051}(10171,\cdot)\)
\(\chi_{40051}(11624,\cdot)\)
\(\chi_{40051}(11679,\cdot)\)
\(\chi_{40051}(12602,\cdot)\)
\(\chi_{40051}(12701,\cdot)\)
\(\chi_{40051}(14252,\cdot)\)
\(\chi_{40051}(14258,\cdot)\)
\(\chi_{40051}(14561,\cdot)\)
\(\chi_{40051}(14867,\cdot)\)
\(\chi_{40051}(14966,\cdot)\)
...
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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 };
\((11255,17546)\) → \((e\left(\frac{3}{110}\right),e\left(\frac{73}{165}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 40051 }(11624, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{53}{55}\right)\) |
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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)
