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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(138787, base_ring=CyclotomicField(1980))
M = H._module
chi = DirichletCharacter(H, M([1512,528,1595]))
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gp:[g,chi] = znchar(Mod(764, 138787))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("138787.764");
| Modulus: | \(138787\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(138787\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1980\) |
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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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\(\chi_{138787}(422,\cdot)\)
\(\chi_{138787}(720,\cdot)\)
\(\chi_{138787}(764,\cdot)\)
\(\chi_{138787}(927,\cdot)\)
\(\chi_{138787}(1105,\cdot)\)
\(\chi_{138787}(1445,\cdot)\)
\(\chi_{138787}(1467,\cdot)\)
\(\chi_{138787}(1626,\cdot)\)
\(\chi_{138787}(1808,\cdot)\)
\(\chi_{138787}(2128,\cdot)\)
\(\chi_{138787}(2425,\cdot)\)
\(\chi_{138787}(2649,\cdot)\)
\(\chi_{138787}(2831,\cdot)\)
\(\chi_{138787}(3789,\cdot)\)
\(\chi_{138787}(4013,\cdot)\)
\(\chi_{138787}(4812,\cdot)\)
\(\chi_{138787}(5124,\cdot)\)
\(\chi_{138787}(5718,\cdot)\)
\(\chi_{138787}(6147,\cdot)\)
\(\chi_{138787}(6488,\cdot)\)
\(\chi_{138787}(7065,\cdot)\)
\(\chi_{138787}(7082,\cdot)\)
\(\chi_{138787}(7923,\cdot)\)
\(\chi_{138787}(8088,\cdot)\)
\(\chi_{138787}(8105,\cdot)\)
\(\chi_{138787}(8264,\cdot)\)
\(\chi_{138787}(8534,\cdot)\)
\(\chi_{138787}(8606,\cdot)\)
\(\chi_{138787}(8875,\cdot)\)
\(\chi_{138787}(9452,\cdot)\)
...
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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 };
\((55057,107449,30009)\) → \((e\left(\frac{42}{55}\right),e\left(\frac{4}{15}\right),e\left(\frac{29}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 138787 }(764, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1919}{1980}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{929}{990}\right)\) | \(e\left(\frac{733}{1980}\right)\) | \(e\left(\frac{251}{660}\right)\) | \(e\left(\frac{292}{495}\right)\) | \(e\left(\frac{599}{660}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{173}{495}\right)\) |
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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)
