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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(277984, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,54,120,27,116]))
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gp:[g,chi] = znchar(Mod(61, 277984))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("277984.61");
| Modulus: | \(277984\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(277984\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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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\(\chi_{277984}(61,\cdot)\)
\(\chi_{277984}(5717,\cdot)\)
\(\chi_{277984}(17685,\cdot)\)
\(\chi_{277984}(22741,\cdot)\)
\(\chi_{277984}(29965,\cdot)\)
\(\chi_{277984}(36725,\cdot)\)
\(\chi_{277984}(38141,\cdot)\)
\(\chi_{277984}(53357,\cdot)\)
\(\chi_{277984}(60821,\cdot)\)
\(\chi_{277984}(61533,\cdot)\)
\(\chi_{277984}(66405,\cdot)\)
\(\chi_{277984}(72397,\cdot)\)
\(\chi_{277984}(72733,\cdot)\)
\(\chi_{277984}(80573,\cdot)\)
\(\chi_{277984}(80909,\cdot)\)
\(\chi_{277984}(87477,\cdot)\)
\(\chi_{277984}(96325,\cdot)\)
\(\chi_{277984}(99445,\cdot)\)
\(\chi_{277984}(115461,\cdot)\)
\(\chi_{277984}(118485,\cdot)\)
\(\chi_{277984}(134405,\cdot)\)
\(\chi_{277984}(136925,\cdot)\)
\(\chi_{277984}(145101,\cdot)\)
\(\chi_{277984}(145381,\cdot)\)
\(\chi_{277984}(161061,\cdot)\)
\(\chi_{277984}(163581,\cdot)\)
\(\chi_{277984}(171757,\cdot)\)
\(\chi_{277984}(173029,\cdot)\)
\(\chi_{277984}(179021,\cdot)\)
\(\chi_{277984}(183461,\cdot)\)
...
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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: |
$\Q(\zeta_{144})$ |
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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 144 polynomial (not computed) |
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sage:chi.fixed_field()
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\((17375,243237,79425,261633,186593)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{3}{16}\right),e\left(\frac{29}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 277984 }(61, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{119}{144}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{113}{144}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{15}{16}\right)\) |
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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)
