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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8370, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([80,45,102]))
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gp:[g,chi] = znchar(Mod(1417, 8370))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8370.1417");
| Modulus: | \(8370\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4185\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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 \(\chi_{4185}(1417,\cdot)\) |
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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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\(\chi_{8370}(13,\cdot)\)
\(\chi_{8370}(43,\cdot)\)
\(\chi_{8370}(313,\cdot)\)
\(\chi_{8370}(517,\cdot)\)
\(\chi_{8370}(637,\cdot)\)
\(\chi_{8370}(1003,\cdot)\)
\(\chi_{8370}(1357,\cdot)\)
\(\chi_{8370}(1417,\cdot)\)
\(\chi_{8370}(1633,\cdot)\)
\(\chi_{8370}(1687,\cdot)\)
\(\chi_{8370}(1717,\cdot)\)
\(\chi_{8370}(1753,\cdot)\)
\(\chi_{8370}(1987,\cdot)\)
\(\chi_{8370}(2473,\cdot)\)
\(\chi_{8370}(2533,\cdot)\)
\(\chi_{8370}(2677,\cdot)\)
\(\chi_{8370}(2803,\cdot)\)
\(\chi_{8370}(2833,\cdot)\)
\(\chi_{8370}(3103,\cdot)\)
\(\chi_{8370}(3307,\cdot)\)
\(\chi_{8370}(3427,\cdot)\)
\(\chi_{8370}(3793,\cdot)\)
\(\chi_{8370}(4147,\cdot)\)
\(\chi_{8370}(4207,\cdot)\)
\(\chi_{8370}(4423,\cdot)\)
\(\chi_{8370}(4477,\cdot)\)
\(\chi_{8370}(4507,\cdot)\)
\(\chi_{8370}(4543,\cdot)\)
\(\chi_{8370}(4777,\cdot)\)
\(\chi_{8370}(5263,\cdot)\)
...
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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 };
\((7751,6697,4591)\) → \((e\left(\frac{4}{9}\right),i,e\left(\frac{17}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8370 }(1417, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{180}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{97}{180}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{169}{180}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{53}{180}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
