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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,165,275,72]))
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gp:[g,chi] = znchar(Mod(203, 8712))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8712.203");
| Modulus: | \(8712\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8712\) |
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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: | 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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\(\chi_{8712}(59,\cdot)\)
\(\chi_{8712}(203,\cdot)\)
\(\chi_{8712}(443,\cdot)\)
\(\chi_{8712}(515,\cdot)\)
\(\chi_{8712}(587,\cdot)\)
\(\chi_{8712}(707,\cdot)\)
\(\chi_{8712}(731,\cdot)\)
\(\chi_{8712}(779,\cdot)\)
\(\chi_{8712}(851,\cdot)\)
\(\chi_{8712}(1235,\cdot)\)
\(\chi_{8712}(1307,\cdot)\)
\(\chi_{8712}(1379,\cdot)\)
\(\chi_{8712}(1499,\cdot)\)
\(\chi_{8712}(1523,\cdot)\)
\(\chi_{8712}(1571,\cdot)\)
\(\chi_{8712}(1643,\cdot)\)
\(\chi_{8712}(1787,\cdot)\)
\(\chi_{8712}(2027,\cdot)\)
\(\chi_{8712}(2099,\cdot)\)
\(\chi_{8712}(2171,\cdot)\)
\(\chi_{8712}(2291,\cdot)\)
\(\chi_{8712}(2315,\cdot)\)
\(\chi_{8712}(2363,\cdot)\)
\(\chi_{8712}(2435,\cdot)\)
\(\chi_{8712}(2579,\cdot)\)
\(\chi_{8712}(2819,\cdot)\)
\(\chi_{8712}(2891,\cdot)\)
\(\chi_{8712}(2963,\cdot)\)
\(\chi_{8712}(3083,\cdot)\)
\(\chi_{8712}(3107,\cdot)\)
...
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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: |
$\Q(\zeta_{165})$ |
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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 330 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((6535,4357,1937,5689)\) → \((-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{12}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8712 }(203, a) \) |
\(1\) | \(1\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{19}{110}\right)\) |
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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)
