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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([72,90,96,63,110]))
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gp:[g,chi] = znchar(Mod(11, 277984))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("277984.11");
| 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: | odd |
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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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\(\chi_{277984}(11,\cdot)\)
\(\chi_{277984}(555,\cdot)\)
\(\chi_{277984}(11435,\cdot)\)
\(\chi_{277984}(12371,\cdot)\)
\(\chi_{277984}(20771,\cdot)\)
\(\chi_{277984}(24483,\cdot)\)
\(\chi_{277984}(27771,\cdot)\)
\(\chi_{277984}(28291,\cdot)\)
\(\chi_{277984}(32883,\cdot)\)
\(\chi_{277984}(36003,\cdot)\)
\(\chi_{277984}(36691,\cdot)\)
\(\chi_{277984}(43075,\cdot)\)
\(\chi_{277984}(46251,\cdot)\)
\(\chi_{277984}(46267,\cdot)\)
\(\chi_{277984}(46587,\cdot)\)
\(\chi_{277984}(52467,\cdot)\)
\(\chi_{277984}(67155,\cdot)\)
\(\chi_{277984}(73467,\cdot)\)
\(\chi_{277984}(73539,\cdot)\)
\(\chi_{277984}(77347,\cdot)\)
\(\chi_{277984}(88491,\cdot)\)
\(\chi_{277984}(92843,\cdot)\)
\(\chi_{277984}(106971,\cdot)\)
\(\chi_{277984}(114643,\cdot)\)
\(\chi_{277984}(118395,\cdot)\)
\(\chi_{277984}(118451,\cdot)\)
\(\chi_{277984}(127843,\cdot)\)
\(\chi_{277984}(134227,\cdot)\)
\(\chi_{277984}(148915,\cdot)\)
\(\chi_{277984}(161051,\cdot)\)
...
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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()
|
\((17375,243237,79425,261633,186593)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{2}{3}\right),e\left(\frac{7}{16}\right),e\left(\frac{55}{72}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 277984 }(11, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{131}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{3}{16}\right)\) |
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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)
