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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5625, base_ring=CyclotomicField(750))
M = H._module
chi = DirichletCharacter(H, M([250,366]))
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gp:[g,chi] = znchar(Mod(166, 5625))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5625.166");
| Modulus: | \(5625\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5625\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(375\) |
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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);
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| 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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\(\chi_{5625}(16,\cdot)\)
\(\chi_{5625}(31,\cdot)\)
\(\chi_{5625}(61,\cdot)\)
\(\chi_{5625}(106,\cdot)\)
\(\chi_{5625}(121,\cdot)\)
\(\chi_{5625}(166,\cdot)\)
\(\chi_{5625}(196,\cdot)\)
\(\chi_{5625}(211,\cdot)\)
\(\chi_{5625}(241,\cdot)\)
\(\chi_{5625}(256,\cdot)\)
\(\chi_{5625}(286,\cdot)\)
\(\chi_{5625}(331,\cdot)\)
\(\chi_{5625}(346,\cdot)\)
\(\chi_{5625}(391,\cdot)\)
\(\chi_{5625}(421,\cdot)\)
\(\chi_{5625}(436,\cdot)\)
\(\chi_{5625}(466,\cdot)\)
\(\chi_{5625}(481,\cdot)\)
\(\chi_{5625}(511,\cdot)\)
\(\chi_{5625}(556,\cdot)\)
\(\chi_{5625}(571,\cdot)\)
\(\chi_{5625}(616,\cdot)\)
\(\chi_{5625}(646,\cdot)\)
\(\chi_{5625}(661,\cdot)\)
\(\chi_{5625}(691,\cdot)\)
\(\chi_{5625}(706,\cdot)\)
\(\chi_{5625}(736,\cdot)\)
\(\chi_{5625}(781,\cdot)\)
\(\chi_{5625}(796,\cdot)\)
\(\chi_{5625}(841,\cdot)\)
...
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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: |
$\Q(\zeta_{375})$ |
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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));
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| Fixed field: |
Number field defined by a degree 375 polynomial (not computed) |
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sage:chi.fixed_field()
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\((4376,1252)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{61}{125}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5625 }(166, a) \) |
\(1\) | \(1\) | \(e\left(\frac{308}{375}\right)\) | \(e\left(\frac{241}{375}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{233}{375}\right)\) | \(e\left(\frac{187}{375}\right)\) | \(e\left(\frac{313}{375}\right)\) | \(e\left(\frac{107}{375}\right)\) | \(e\left(\frac{53}{125}\right)\) | \(e\left(\frac{123}{125}\right)\) |
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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)
