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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39325, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([33,16,55]))
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gp:[g,chi] = znchar(Mod(7214, 39325))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39325.7214");
| Modulus: | \(39325\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(39325\) |
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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: | 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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\(\chi_{39325}(64,\cdot)\)
\(\chi_{39325}(1104,\cdot)\)
\(\chi_{39325}(1169,\cdot)\)
\(\chi_{39325}(3184,\cdot)\)
\(\chi_{39325}(4744,\cdot)\)
\(\chi_{39325}(6759,\cdot)\)
\(\chi_{39325}(7214,\cdot)\)
\(\chi_{39325}(8254,\cdot)\)
\(\chi_{39325}(8319,\cdot)\)
\(\chi_{39325}(10334,\cdot)\)
\(\chi_{39325}(10789,\cdot)\)
\(\chi_{39325}(11829,\cdot)\)
\(\chi_{39325}(11894,\cdot)\)
\(\chi_{39325}(13909,\cdot)\)
\(\chi_{39325}(14364,\cdot)\)
\(\chi_{39325}(15404,\cdot)\)
\(\chi_{39325}(15469,\cdot)\)
\(\chi_{39325}(17484,\cdot)\)
\(\chi_{39325}(17939,\cdot)\)
\(\chi_{39325}(18979,\cdot)\)
\(\chi_{39325}(19044,\cdot)\)
\(\chi_{39325}(21059,\cdot)\)
\(\chi_{39325}(21514,\cdot)\)
\(\chi_{39325}(22554,\cdot)\)
\(\chi_{39325}(22619,\cdot)\)
\(\chi_{39325}(24634,\cdot)\)
\(\chi_{39325}(25089,\cdot)\)
\(\chi_{39325}(26129,\cdot)\)
\(\chi_{39325}(26194,\cdot)\)
\(\chi_{39325}(28209,\cdot)\)
...
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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()
|
\((18877,11376,9076)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{8}{55}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 39325 }(7214, a) \) |
\(1\) | \(1\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) |
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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)
