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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(93345, base_ring=CyclotomicField(252))
M = H._module
chi = DirichletCharacter(H, M([0,189,90,32]))
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gp:[g,chi] = znchar(Mod(3373, 93345))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("93345.3373");
| Modulus: | \(93345\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(31115\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(252\) |
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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 \(\chi_{31115}(3373,\cdot)\) |
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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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\(\chi_{93345}(13,\cdot)\)
\(\chi_{93345}(1693,\cdot)\)
\(\chi_{93345}(3373,\cdot)\)
\(\chi_{93345}(4108,\cdot)\)
\(\chi_{93345}(6082,\cdot)\)
\(\chi_{93345}(8413,\cdot)\)
\(\chi_{93345}(8518,\cdot)\)
\(\chi_{93345}(10933,\cdot)\)
\(\chi_{93345}(13663,\cdot)\)
\(\chi_{93345}(15742,\cdot)\)
\(\chi_{93345}(15763,\cdot)\)
\(\chi_{93345}(16582,\cdot)\)
\(\chi_{93345}(17233,\cdot)\)
\(\chi_{93345}(18367,\cdot)\)
\(\chi_{93345}(18682,\cdot)\)
\(\chi_{93345}(20362,\cdot)\)
\(\chi_{93345}(20698,\cdot)\)
\(\chi_{93345}(22042,\cdot)\)
\(\chi_{93345}(22168,\cdot)\)
\(\chi_{93345}(22777,\cdot)\)
\(\chi_{93345}(23218,\cdot)\)
\(\chi_{93345}(26053,\cdot)\)
\(\chi_{93345}(27082,\cdot)\)
\(\chi_{93345}(27187,\cdot)\)
\(\chi_{93345}(29602,\cdot)\)
\(\chi_{93345}(32332,\cdot)\)
\(\chi_{93345}(33823,\cdot)\)
\(\chi_{93345}(34432,\cdot)\)
\(\chi_{93345}(35902,\cdot)\)
\(\chi_{93345}(36343,\cdot)\)
...
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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 };
\((62231,74677,30481,69091)\) → \((1,-i,e\left(\frac{5}{14}\right),e\left(\frac{8}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 93345 }(3373, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{127}{252}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{252}\right)\) | \(e\left(\frac{47}{252}\right)\) |
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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)
