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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87379, base_ring=CyclotomicField(370))
M = H._module
chi = DirichletCharacter(H, M([0,277]))
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gp:[g,chi] = znchar(Mod(6963, 87379))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("87379.6963");
| Modulus: | \(87379\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1481\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(370\) |
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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 \(\chi_{1481}(1039,\cdot)\) |
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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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\(\chi_{87379}(119,\cdot)\)
\(\chi_{87379}(178,\cdot)\)
\(\chi_{87379}(237,\cdot)\)
\(\chi_{87379}(414,\cdot)\)
\(\chi_{87379}(1417,\cdot)\)
\(\chi_{87379}(2420,\cdot)\)
\(\chi_{87379}(2833,\cdot)\)
\(\chi_{87379}(3777,\cdot)\)
\(\chi_{87379}(6196,\cdot)\)
\(\chi_{87379}(6491,\cdot)\)
\(\chi_{87379}(6668,\cdot)\)
\(\chi_{87379}(6963,\cdot)\)
\(\chi_{87379}(7730,\cdot)\)
\(\chi_{87379}(7966,\cdot)\)
\(\chi_{87379}(8261,\cdot)\)
\(\chi_{87379}(8438,\cdot)\)
\(\chi_{87379}(9618,\cdot)\)
\(\chi_{87379}(9736,\cdot)\)
\(\chi_{87379}(10149,\cdot)\)
\(\chi_{87379}(11152,\cdot)\)
\(\chi_{87379}(11211,\cdot)\)
\(\chi_{87379}(13217,\cdot)\)
\(\chi_{87379}(13512,\cdot)\)
\(\chi_{87379}(13689,\cdot)\)
\(\chi_{87379}(15459,\cdot)\)
\(\chi_{87379}(15695,\cdot)\)
\(\chi_{87379}(16639,\cdot)\)
\(\chi_{87379}(17052,\cdot)\)
\(\chi_{87379}(17347,\cdot)\)
\(\chi_{87379}(17406,\cdot)\)
...
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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 };
\((14811,57762)\) → \((1,e\left(\frac{277}{370}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 87379 }(6963, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{185}\right)\) | \(e\left(\frac{277}{370}\right)\) | \(e\left(\frac{106}{185}\right)\) | \(e\left(\frac{183}{185}\right)\) | \(e\left(\frac{13}{370}\right)\) | \(e\left(\frac{29}{185}\right)\) | \(e\left(\frac{159}{185}\right)\) | \(e\left(\frac{92}{185}\right)\) | \(e\left(\frac{51}{185}\right)\) | \(e\left(\frac{25}{74}\right)\) |
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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)
