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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([84,10]))
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gp:[g,chi] = znchar(Mod(156, 1225))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1225.156");
| Modulus: | \(1225\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1225\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(105\) |
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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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\(\chi_{1225}(11,\cdot)\)
\(\chi_{1225}(16,\cdot)\)
\(\chi_{1225}(46,\cdot)\)
\(\chi_{1225}(81,\cdot)\)
\(\chi_{1225}(86,\cdot)\)
\(\chi_{1225}(121,\cdot)\)
\(\chi_{1225}(156,\cdot)\)
\(\chi_{1225}(186,\cdot)\)
\(\chi_{1225}(191,\cdot)\)
\(\chi_{1225}(221,\cdot)\)
\(\chi_{1225}(256,\cdot)\)
\(\chi_{1225}(261,\cdot)\)
\(\chi_{1225}(291,\cdot)\)
\(\chi_{1225}(296,\cdot)\)
\(\chi_{1225}(331,\cdot)\)
\(\chi_{1225}(366,\cdot)\)
\(\chi_{1225}(396,\cdot)\)
\(\chi_{1225}(431,\cdot)\)
\(\chi_{1225}(436,\cdot)\)
\(\chi_{1225}(466,\cdot)\)
\(\chi_{1225}(506,\cdot)\)
\(\chi_{1225}(536,\cdot)\)
\(\chi_{1225}(541,\cdot)\)
\(\chi_{1225}(571,\cdot)\)
\(\chi_{1225}(611,\cdot)\)
\(\chi_{1225}(641,\cdot)\)
\(\chi_{1225}(646,\cdot)\)
\(\chi_{1225}(681,\cdot)\)
\(\chi_{1225}(711,\cdot)\)
\(\chi_{1225}(746,\cdot)\)
...
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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: |
$\Q(\zeta_{105})$ |
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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 105 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1177,101)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{1}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(156, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{58}{105}\right)\) |
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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)
