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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104000, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,105,42,10]))
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gp:[g,chi] = znchar(Mod(78743, 104000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104000.78743");
| 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: | \(10400\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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 \(\chi_{10400}(3603,\cdot)\) |
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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);
|
\(\chi_{104000}(7,\cdot)\)
\(\chi_{104000}(6407,\cdot)\)
\(\chi_{104000}(10407,\cdot)\)
\(\chi_{104000}(12343,\cdot)\)
\(\chi_{104000}(16343,\cdot)\)
\(\chi_{104000}(22743,\cdot)\)
\(\chi_{104000}(26743,\cdot)\)
\(\chi_{104000}(27207,\cdot)\)
\(\chi_{104000}(31207,\cdot)\)
\(\chi_{104000}(33143,\cdot)\)
\(\chi_{104000}(37143,\cdot)\)
\(\chi_{104000}(37607,\cdot)\)
\(\chi_{104000}(41607,\cdot)\)
\(\chi_{104000}(43543,\cdot)\)
\(\chi_{104000}(47543,\cdot)\)
\(\chi_{104000}(48007,\cdot)\)
\(\chi_{104000}(52007,\cdot)\)
\(\chi_{104000}(58407,\cdot)\)
\(\chi_{104000}(62407,\cdot)\)
\(\chi_{104000}(64343,\cdot)\)
\(\chi_{104000}(68343,\cdot)\)
\(\chi_{104000}(74743,\cdot)\)
\(\chi_{104000}(78743,\cdot)\)
\(\chi_{104000}(79207,\cdot)\)
\(\chi_{104000}(83207,\cdot)\)
\(\chi_{104000}(85143,\cdot)\)
\(\chi_{104000}(89143,\cdot)\)
\(\chi_{104000}(89607,\cdot)\)
\(\chi_{104000}(93607,\cdot)\)
\(\chi_{104000}(95543,\cdot)\)
...
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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: |
$\Q(\zeta_{120})$ |
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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 120 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,58501,77377,64001)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{7}{20}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 104000 }(78743, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{79}{120}\right)\) |
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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)
