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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(97693, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([418,252]))
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gp:[g,chi] = znchar(Mod(15504, 97693))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("97693.15504");
| 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: | \(231\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{97693}(1444,\cdot)\)
\(\chi_{97693}(1867,\cdot)\)
\(\chi_{97693}(2077,\cdot)\)
\(\chi_{97693}(3115,\cdot)\)
\(\chi_{97693}(3134,\cdot)\)
\(\chi_{97693}(4182,\cdot)\)
\(\chi_{97693}(4504,\cdot)\)
\(\chi_{97693}(4592,\cdot)\)
\(\chi_{97693}(4685,\cdot)\)
\(\chi_{97693}(5318,\cdot)\)
\(\chi_{97693}(5455,\cdot)\)
\(\chi_{97693}(5981,\cdot)\)
\(\chi_{97693}(6153,\cdot)\)
\(\chi_{97693}(8812,\cdot)\)
\(\chi_{97693}(10201,\cdot)\)
\(\chi_{97693}(12838,\cdot)\)
\(\chi_{97693}(14315,\cdot)\)
\(\chi_{97693}(14871,\cdot)\)
\(\chi_{97693}(14950,\cdot)\)
\(\chi_{97693}(15437,\cdot)\)
\(\chi_{97693}(15504,\cdot)\)
\(\chi_{97693}(16079,\cdot)\)
\(\chi_{97693}(16723,\cdot)\)
\(\chi_{97693}(17356,\cdot)\)
\(\chi_{97693}(17487,\cdot)\)
\(\chi_{97693}(17556,\cdot)\)
\(\chi_{97693}(17841,\cdot)\)
\(\chi_{97693}(18535,\cdot)\)
\(\chi_{97693}(19802,\cdot)\)
\(\chi_{97693}(19808,\cdot)\)
...
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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 };
\((81026,33339)\) → \((e\left(\frac{19}{21}\right),e\left(\frac{6}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 97693 }(15504, a) \) |
\(1\) | \(1\) | \(e\left(\frac{104}{231}\right)\) | \(e\left(\frac{104}{231}\right)\) | \(e\left(\frac{208}{231}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{208}{231}\right)\) | \(e\left(\frac{50}{231}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{208}{231}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{2}{77}\right)\) |
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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)
