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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21511, base_ring=CyclotomicField(438))
M = H._module
chi = DirichletCharacter(H, M([0,149]))
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gp:[g,chi] = znchar(Mod(736, 21511))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21511.736");
| Modulus: | \(21511\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(439\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(438\) |
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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 \(\chi_{439}(297,\cdot)\) |
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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);
|
\(\chi_{21511}(148,\cdot)\)
\(\chi_{21511}(197,\cdot)\)
\(\chi_{21511}(344,\cdot)\)
\(\chi_{21511}(540,\cdot)\)
\(\chi_{21511}(589,\cdot)\)
\(\chi_{21511}(687,\cdot)\)
\(\chi_{21511}(736,\cdot)\)
\(\chi_{21511}(785,\cdot)\)
\(\chi_{21511}(834,\cdot)\)
\(\chi_{21511}(1177,\cdot)\)
\(\chi_{21511}(1226,\cdot)\)
\(\chi_{21511}(1422,\cdot)\)
\(\chi_{21511}(1618,\cdot)\)
\(\chi_{21511}(1716,\cdot)\)
\(\chi_{21511}(1863,\cdot)\)
\(\chi_{21511}(1912,\cdot)\)
\(\chi_{21511}(1961,\cdot)\)
\(\chi_{21511}(2157,\cdot)\)
\(\chi_{21511}(2255,\cdot)\)
\(\chi_{21511}(2353,\cdot)\)
\(\chi_{21511}(2402,\cdot)\)
\(\chi_{21511}(2696,\cdot)\)
\(\chi_{21511}(2990,\cdot)\)
\(\chi_{21511}(3088,\cdot)\)
\(\chi_{21511}(3529,\cdot)\)
\(\chi_{21511}(3578,\cdot)\)
\(\chi_{21511}(3676,\cdot)\)
\(\chi_{21511}(3725,\cdot)\)
\(\chi_{21511}(3774,\cdot)\)
\(\chi_{21511}(4019,\cdot)\)
...
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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 };
\((21073,3088)\) → \((1,e\left(\frac{149}{438}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 21511 }(736, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{39}{73}\right)\) | \(e\left(\frac{23}{146}\right)\) | \(e\left(\frac{5}{73}\right)\) | \(e\left(\frac{40}{219}\right)\) | \(e\left(\frac{101}{146}\right)\) | \(e\left(\frac{44}{73}\right)\) | \(e\left(\frac{23}{73}\right)\) | \(e\left(\frac{157}{219}\right)\) | \(e\left(\frac{47}{219}\right)\) | \(e\left(\frac{33}{146}\right)\) |
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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)
