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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104000, base_ring=CyclotomicField(600))
M = H._module
chi = DirichletCharacter(H, M([300,375,432,250]))
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gp:[g,chi] = znchar(Mod(71, 104000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104000.71");
| Modulus: | \(104000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(52000\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(600\) |
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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 \(\chi_{52000}(6571,\cdot)\) |
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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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\(\chi_{104000}(71,\cdot)\)
\(\chi_{104000}(631,\cdot)\)
\(\chi_{104000}(791,\cdot)\)
\(\chi_{104000}(1991,\cdot)\)
\(\chi_{104000}(2711,\cdot)\)
\(\chi_{104000}(2871,\cdot)\)
\(\chi_{104000}(4071,\cdot)\)
\(\chi_{104000}(4231,\cdot)\)
\(\chi_{104000}(4791,\cdot)\)
\(\chi_{104000}(6311,\cdot)\)
\(\chi_{104000}(6871,\cdot)\)
\(\chi_{104000}(7031,\cdot)\)
\(\chi_{104000}(8231,\cdot)\)
\(\chi_{104000}(8391,\cdot)\)
\(\chi_{104000}(9111,\cdot)\)
\(\chi_{104000}(10311,\cdot)\)
\(\chi_{104000}(10471,\cdot)\)
\(\chi_{104000}(11031,\cdot)\)
\(\chi_{104000}(11191,\cdot)\)
\(\chi_{104000}(12391,\cdot)\)
\(\chi_{104000}(13111,\cdot)\)
\(\chi_{104000}(13271,\cdot)\)
\(\chi_{104000}(14471,\cdot)\)
\(\chi_{104000}(14631,\cdot)\)
\(\chi_{104000}(15191,\cdot)\)
\(\chi_{104000}(16711,\cdot)\)
\(\chi_{104000}(17271,\cdot)\)
\(\chi_{104000}(17431,\cdot)\)
\(\chi_{104000}(18631,\cdot)\)
\(\chi_{104000}(18791,\cdot)\)
...
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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: |
$\Q(\zeta_{600})$ |
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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 600 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,58501,77377,64001)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{18}{25}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 104000 }(71, a) \) |
\(1\) | \(1\) | \(e\left(\frac{49}{600}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{300}\right)\) | \(e\left(\frac{157}{600}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{551}{600}\right)\) | \(e\left(\frac{123}{200}\right)\) | \(e\left(\frac{221}{300}\right)\) | \(e\left(\frac{49}{200}\right)\) | \(e\left(\frac{109}{600}\right)\) |
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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)
