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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4356, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([0,55,159]))
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gp:[g,chi] = znchar(Mod(1361, 4356))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4356.1361");
| Modulus: | \(4356\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1089\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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 \(\chi_{1089}(272,\cdot)\) |
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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);
|
\(\chi_{4356}(29,\cdot)\)
\(\chi_{4356}(41,\cdot)\)
\(\chi_{4356}(101,\cdot)\)
\(\chi_{4356}(149,\cdot)\)
\(\chi_{4356}(173,\cdot)\)
\(\chi_{4356}(281,\cdot)\)
\(\chi_{4356}(293,\cdot)\)
\(\chi_{4356}(365,\cdot)\)
\(\chi_{4356}(425,\cdot)\)
\(\chi_{4356}(437,\cdot)\)
\(\chi_{4356}(497,\cdot)\)
\(\chi_{4356}(545,\cdot)\)
\(\chi_{4356}(569,\cdot)\)
\(\chi_{4356}(677,\cdot)\)
\(\chi_{4356}(689,\cdot)\)
\(\chi_{4356}(761,\cdot)\)
\(\chi_{4356}(821,\cdot)\)
\(\chi_{4356}(833,\cdot)\)
\(\chi_{4356}(893,\cdot)\)
\(\chi_{4356}(1073,\cdot)\)
\(\chi_{4356}(1085,\cdot)\)
\(\chi_{4356}(1157,\cdot)\)
\(\chi_{4356}(1217,\cdot)\)
\(\chi_{4356}(1229,\cdot)\)
\(\chi_{4356}(1289,\cdot)\)
\(\chi_{4356}(1337,\cdot)\)
\(\chi_{4356}(1361,\cdot)\)
\(\chi_{4356}(1469,\cdot)\)
\(\chi_{4356}(1481,\cdot)\)
\(\chi_{4356}(1553,\cdot)\)
...
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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 };
\((2179,1937,1333)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{53}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4356 }(1361, a) \) |
\(1\) | \(1\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{13}{330}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{29}{55}\right)\) |
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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)
