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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11809, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([110,273]))
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gp:[g,chi] = znchar(Mod(698, 11809))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11809.698");
| Modulus: | \(11809\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(11809\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(420\) |
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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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\(\chi_{11809}(40,\cdot)\)
\(\chi_{11809}(201,\cdot)\)
\(\chi_{11809}(376,\cdot)\)
\(\chi_{11809}(488,\cdot)\)
\(\chi_{11809}(507,\cdot)\)
\(\chi_{11809}(698,\cdot)\)
\(\chi_{11809}(829,\cdot)\)
\(\chi_{11809}(1004,\cdot)\)
\(\chi_{11809}(1165,\cdot)\)
\(\chi_{11809}(1230,\cdot)\)
\(\chi_{11809}(1340,\cdot)\)
\(\chi_{11809}(1552,\cdot)\)
\(\chi_{11809}(1662,\cdot)\)
\(\chi_{11809}(1727,\cdot)\)
\(\chi_{11809}(1888,\cdot)\)
\(\chi_{11809}(2063,\cdot)\)
\(\chi_{11809}(2194,\cdot)\)
\(\chi_{11809}(2385,\cdot)\)
\(\chi_{11809}(2404,\cdot)\)
\(\chi_{11809}(2516,\cdot)\)
\(\chi_{11809}(2691,\cdot)\)
\(\chi_{11809}(2852,\cdot)\)
\(\chi_{11809}(2917,\cdot)\)
\(\chi_{11809}(3027,\cdot)\)
\(\chi_{11809}(3127,\cdot)\)
\(\chi_{11809}(3139,\cdot)\)
\(\chi_{11809}(3239,\cdot)\)
\(\chi_{11809}(3349,\cdot)\)
\(\chi_{11809}(3414,\cdot)\)
\(\chi_{11809}(3575,\cdot)\)
...
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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 };
\((5785,6273)\) → \((e\left(\frac{11}{42}\right),e\left(\frac{13}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 11809 }(698, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{127}{210}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{19}{105}\right)\) |
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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)
