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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104000, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([150,0,201,275]))
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gp:[g,chi] = znchar(Mod(10303, 104000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104000.10303");
| 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: | \(6500\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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 \(\chi_{6500}(3803,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{104000}(63,\cdot)\)
\(\chi_{104000}(1983,\cdot)\)
\(\chi_{104000}(2047,\cdot)\)
\(\chi_{104000}(3967,\cdot)\)
\(\chi_{104000}(4223,\cdot)\)
\(\chi_{104000}(8127,\cdot)\)
\(\chi_{104000}(8383,\cdot)\)
\(\chi_{104000}(10303,\cdot)\)
\(\chi_{104000}(10367,\cdot)\)
\(\chi_{104000}(12287,\cdot)\)
\(\chi_{104000}(14463,\cdot)\)
\(\chi_{104000}(14527,\cdot)\)
\(\chi_{104000}(16447,\cdot)\)
\(\chi_{104000}(16703,\cdot)\)
\(\chi_{104000}(18623,\cdot)\)
\(\chi_{104000}(18687,\cdot)\)
\(\chi_{104000}(20863,\cdot)\)
\(\chi_{104000}(22783,\cdot)\)
\(\chi_{104000}(22847,\cdot)\)
\(\chi_{104000}(24767,\cdot)\)
\(\chi_{104000}(25023,\cdot)\)
\(\chi_{104000}(28927,\cdot)\)
\(\chi_{104000}(29183,\cdot)\)
\(\chi_{104000}(31103,\cdot)\)
\(\chi_{104000}(31167,\cdot)\)
\(\chi_{104000}(33087,\cdot)\)
\(\chi_{104000}(35263,\cdot)\)
\(\chi_{104000}(35327,\cdot)\)
\(\chi_{104000}(37247,\cdot)\)
\(\chi_{104000}(37503,\cdot)\)
...
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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: |
$\Q(\zeta_{300})$ |
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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 300 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,58501,77377,64001)\) → \((-1,1,e\left(\frac{67}{100}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 104000 }(10303, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{257}{300}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{251}{300}\right)\) | \(e\left(\frac{223}{300}\right)\) | \(e\left(\frac{43}{300}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{131}{300}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{31}{150}\right)\) |
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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)
