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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(228672, base_ring=CyclotomicField(396))
M = H._module
chi = DirichletCharacter(H, M([198,297,66,376]))
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gp:[g,chi] = znchar(Mod(47, 228672))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("228672.47");
| Modulus: | \(228672\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(57168\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(396\) |
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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 \(\chi_{57168}(14339,\cdot)\) |
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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: | no |
| 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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\(\chi_{228672}(47,\cdot)\)
\(\chi_{228672}(3791,\cdot)\)
\(\chi_{228672}(4559,\cdot)\)
\(\chi_{228672}(6671,\cdot)\)
\(\chi_{228672}(7439,\cdot)\)
\(\chi_{228672}(9263,\cdot)\)
\(\chi_{228672}(10319,\cdot)\)
\(\chi_{228672}(10895,\cdot)\)
\(\chi_{228672}(11183,\cdot)\)
\(\chi_{228672}(15215,\cdot)\)
\(\chi_{228672}(16367,\cdot)\)
\(\chi_{228672}(18671,\cdot)\)
\(\chi_{228672}(19631,\cdot)\)
\(\chi_{228672}(19919,\cdot)\)
\(\chi_{228672}(20207,\cdot)\)
\(\chi_{228672}(21647,\cdot)\)
\(\chi_{228672}(22511,\cdot)\)
\(\chi_{228672}(24527,\cdot)\)
\(\chi_{228672}(24719,\cdot)\)
\(\chi_{228672}(29135,\cdot)\)
\(\chi_{228672}(30191,\cdot)\)
\(\chi_{228672}(31727,\cdot)\)
\(\chi_{228672}(33359,\cdot)\)
\(\chi_{228672}(34223,\cdot)\)
\(\chi_{228672}(37391,\cdot)\)
\(\chi_{228672}(38063,\cdot)\)
\(\chi_{228672}(40847,\cdot)\)
\(\chi_{228672}(41231,\cdot)\)
\(\chi_{228672}(41711,\cdot)\)
\(\chi_{228672}(41807,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{396})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 396 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((21439,185797,25409,169921)\) → \((-1,-i,e\left(\frac{1}{6}\right),e\left(\frac{94}{99}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 228672 }(47, a) \) |
\(1\) | \(1\) | \(e\left(\frac{211}{396}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{137}{396}\right)\) | \(e\left(\frac{191}{396}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{113}{396}\right)\) | \(e\left(\frac{85}{198}\right)\) | \(e\left(\frac{13}{198}\right)\) | \(e\left(\frac{365}{396}\right)\) | \(e\left(\frac{31}{66}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
