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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6795, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([0,225,286]))
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gp:[g,chi] = znchar(Mod(1468, 6795))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6795.1468");
| Modulus: | \(6795\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(755\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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 \(\chi_{755}(713,\cdot)\) |
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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: | even |
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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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\(\chi_{6795}(82,\cdot)\)
\(\chi_{6795}(163,\cdot)\)
\(\chi_{6795}(253,\cdot)\)
\(\chi_{6795}(262,\cdot)\)
\(\chi_{6795}(442,\cdot)\)
\(\chi_{6795}(568,\cdot)\)
\(\chi_{6795}(658,\cdot)\)
\(\chi_{6795}(667,\cdot)\)
\(\chi_{6795}(712,\cdot)\)
\(\chi_{6795}(1018,\cdot)\)
\(\chi_{6795}(1063,\cdot)\)
\(\chi_{6795}(1072,\cdot)\)
\(\chi_{6795}(1108,\cdot)\)
\(\chi_{6795}(1153,\cdot)\)
\(\chi_{6795}(1198,\cdot)\)
\(\chi_{6795}(1243,\cdot)\)
\(\chi_{6795}(1297,\cdot)\)
\(\chi_{6795}(1342,\cdot)\)
\(\chi_{6795}(1468,\cdot)\)
\(\chi_{6795}(1522,\cdot)\)
\(\chi_{6795}(1558,\cdot)\)
\(\chi_{6795}(1603,\cdot)\)
\(\chi_{6795}(1612,\cdot)\)
\(\chi_{6795}(1738,\cdot)\)
\(\chi_{6795}(1873,\cdot)\)
\(\chi_{6795}(1918,\cdot)\)
\(\chi_{6795}(1927,\cdot)\)
\(\chi_{6795}(2017,\cdot)\)
\(\chi_{6795}(2278,\cdot)\)
\(\chi_{6795}(2377,\cdot)\)
...
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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 };
\((6041,5437,3781)\) → \((1,-i,e\left(\frac{143}{150}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 6795 }(1468, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{187}{300}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{101}{300}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{300}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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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)
