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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(97693, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([44,25]))
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gp:[g,chi] = znchar(Mod(18996, 97693))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("97693.18996");
| Modulus: | \(97693\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(97693\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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: | yes |
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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);
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\(\chi_{97693}(2146,\cdot)\)
\(\chi_{97693}(2588,\cdot)\)
\(\chi_{97693}(6164,\cdot)\)
\(\chi_{97693}(6749,\cdot)\)
\(\chi_{97693}(6776,\cdot)\)
\(\chi_{97693}(8144,\cdot)\)
\(\chi_{97693}(13708,\cdot)\)
\(\chi_{97693}(18996,\cdot)\)
\(\chi_{97693}(23913,\cdot)\)
\(\chi_{97693}(24349,\cdot)\)
\(\chi_{97693}(26358,\cdot)\)
\(\chi_{97693}(26691,\cdot)\)
\(\chi_{97693}(27602,\cdot)\)
\(\chi_{97693}(29599,\cdot)\)
\(\chi_{97693}(30858,\cdot)\)
\(\chi_{97693}(32092,\cdot)\)
\(\chi_{97693}(32683,\cdot)\)
\(\chi_{97693}(32887,\cdot)\)
\(\chi_{97693}(33630,\cdot)\)
\(\chi_{97693}(35400,\cdot)\)
\(\chi_{97693}(35914,\cdot)\)
\(\chi_{97693}(42427,\cdot)\)
\(\chi_{97693}(44660,\cdot)\)
\(\chi_{97693}(45519,\cdot)\)
\(\chi_{97693}(50944,\cdot)\)
\(\chi_{97693}(52332,\cdot)\)
\(\chi_{97693}(57234,\cdot)\)
\(\chi_{97693}(58550,\cdot)\)
\(\chi_{97693}(58623,\cdot)\)
\(\chi_{97693}(59549,\cdot)\)
...
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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 };
\((81026,33339)\) → \((e\left(\frac{22}{105}\right),e\left(\frac{5}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 97693 }(18996, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{13}{210}\right)\) |
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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)
