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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1864, base_ring=CyclotomicField(116))
M = H._module
chi = DirichletCharacter(H, M([0,0,27]))
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gp:[g,chi] = znchar(Mod(649, 1864))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1864.649");
| Modulus: | \(1864\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(233\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(116\) |
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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: | no, induced from \(\chi_{233}(183,\cdot)\) |
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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);
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\(\chi_{1864}(9,\cdot)\)
\(\chi_{1864}(25,\cdot)\)
\(\chi_{1864}(33,\cdot)\)
\(\chi_{1864}(113,\cdot)\)
\(\chi_{1864}(121,\cdot)\)
\(\chi_{1864}(129,\cdot)\)
\(\chi_{1864}(161,\cdot)\)
\(\chi_{1864}(177,\cdot)\)
\(\chi_{1864}(289,\cdot)\)
\(\chi_{1864}(305,\cdot)\)
\(\chi_{1864}(337,\cdot)\)
\(\chi_{1864}(345,\cdot)\)
\(\chi_{1864}(353,\cdot)\)
\(\chi_{1864}(433,\cdot)\)
\(\chi_{1864}(441,\cdot)\)
\(\chi_{1864}(457,\cdot)\)
\(\chi_{1864}(473,\cdot)\)
\(\chi_{1864}(481,\cdot)\)
\(\chi_{1864}(497,\cdot)\)
\(\chi_{1864}(521,\cdot)\)
\(\chi_{1864}(633,\cdot)\)
\(\chi_{1864}(649,\cdot)\)
\(\chi_{1864}(673,\cdot)\)
\(\chi_{1864}(681,\cdot)\)
\(\chi_{1864}(713,\cdot)\)
\(\chi_{1864}(729,\cdot)\)
\(\chi_{1864}(761,\cdot)\)
\(\chi_{1864}(809,\cdot)\)
\(\chi_{1864}(945,\cdot)\)
\(\chi_{1864}(1033,\cdot)\)
...
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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 };
\((1399,933,1401)\) → \((1,1,e\left(\frac{27}{116}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1864 }(649, a) \) |
\(1\) | \(1\) | \(e\left(\frac{27}{116}\right)\) | \(e\left(\frac{47}{116}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{99}{116}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{113}{116}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{105}{116}\right)\) |
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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)
