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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(56265, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([0,165,210,286]))
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gp:[g,chi] = znchar(Mod(2287, 56265))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("56265.2287");
| Modulus: | \(56265\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(18755\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(660\) |
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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 \(\chi_{18755}(2287,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{56265}(43,\cdot)\)
\(\chi_{56265}(208,\cdot)\)
\(\chi_{56265}(538,\cdot)\)
\(\chi_{56265}(637,\cdot)\)
\(\chi_{56265}(703,\cdot)\)
\(\chi_{56265}(2287,\cdot)\)
\(\chi_{56265}(2452,\cdot)\)
\(\chi_{56265}(2617,\cdot)\)
\(\chi_{56265}(2683,\cdot)\)
\(\chi_{56265}(3112,\cdot)\)
\(\chi_{56265}(3277,\cdot)\)
\(\chi_{56265}(3607,\cdot)\)
\(\chi_{56265}(3772,\cdot)\)
\(\chi_{56265}(4333,\cdot)\)
\(\chi_{56265}(4498,\cdot)\)
\(\chi_{56265}(4663,\cdot)\)
\(\chi_{56265}(5158,\cdot)\)
\(\chi_{56265}(5653,\cdot)\)
\(\chi_{56265}(5752,\cdot)\)
\(\chi_{56265}(5818,\cdot)\)
\(\chi_{56265}(7402,\cdot)\)
\(\chi_{56265}(7567,\cdot)\)
\(\chi_{56265}(7732,\cdot)\)
\(\chi_{56265}(7798,\cdot)\)
\(\chi_{56265}(8392,\cdot)\)
\(\chi_{56265}(8722,\cdot)\)
\(\chi_{56265}(8887,\cdot)\)
\(\chi_{56265}(9448,\cdot)\)
\(\chi_{56265}(9613,\cdot)\)
\(\chi_{56265}(9778,\cdot)\)
...
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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 };
\((37511,22507,32551,39931)\) → \((1,i,e\left(\frac{7}{22}\right),e\left(\frac{13}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 56265 }(2287, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{403}{660}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{431}{660}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{577}{660}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{159}{220}\right)\) |
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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)
