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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([77,20]))
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gp:[g,chi] = znchar(Mod(809, 1150))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1150.809");
| Modulus: | \(1150\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(575\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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 \(\chi_{575}(234,\cdot)\) |
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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: | 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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\(\chi_{1150}(9,\cdot)\)
\(\chi_{1150}(29,\cdot)\)
\(\chi_{1150}(39,\cdot)\)
\(\chi_{1150}(59,\cdot)\)
\(\chi_{1150}(119,\cdot)\)
\(\chi_{1150}(169,\cdot)\)
\(\chi_{1150}(179,\cdot)\)
\(\chi_{1150}(209,\cdot)\)
\(\chi_{1150}(219,\cdot)\)
\(\chi_{1150}(239,\cdot)\)
\(\chi_{1150}(259,\cdot)\)
\(\chi_{1150}(269,\cdot)\)
\(\chi_{1150}(279,\cdot)\)
\(\chi_{1150}(289,\cdot)\)
\(\chi_{1150}(409,\cdot)\)
\(\chi_{1150}(439,\cdot)\)
\(\chi_{1150}(469,\cdot)\)
\(\chi_{1150}(489,\cdot)\)
\(\chi_{1150}(509,\cdot)\)
\(\chi_{1150}(519,\cdot)\)
\(\chi_{1150}(579,\cdot)\)
\(\chi_{1150}(629,\cdot)\)
\(\chi_{1150}(639,\cdot)\)
\(\chi_{1150}(669,\cdot)\)
\(\chi_{1150}(679,\cdot)\)
\(\chi_{1150}(719,\cdot)\)
\(\chi_{1150}(729,\cdot)\)
\(\chi_{1150}(739,\cdot)\)
\(\chi_{1150}(809,\cdot)\)
\(\chi_{1150}(859,\cdot)\)
...
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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: |
$\Q(\zeta_{55})$ |
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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 110 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((277,51)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 1150 }(809, a) \) |
\(1\) | \(1\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{37}{55}\right)\) |
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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)
