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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12935, base_ring=CyclotomicField(396))
M = H._module
chi = DirichletCharacter(H, M([198,99,142]))
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gp:[g,chi] = znchar(Mod(684, 12935))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12935.684");
| Modulus: | \(12935\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12935\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(396\) |
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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);
|
\(\chi_{12935}(34,\cdot)\)
\(\chi_{12935}(44,\cdot)\)
\(\chi_{12935}(99,\cdot)\)
\(\chi_{12935}(164,\cdot)\)
\(\chi_{12935}(229,\cdot)\)
\(\chi_{12935}(294,\cdot)\)
\(\chi_{12935}(304,\cdot)\)
\(\chi_{12935}(369,\cdot)\)
\(\chi_{12935}(564,\cdot)\)
\(\chi_{12935}(619,\cdot)\)
\(\chi_{12935}(684,\cdot)\)
\(\chi_{12935}(694,\cdot)\)
\(\chi_{12935}(749,\cdot)\)
\(\chi_{12935}(944,\cdot)\)
\(\chi_{12935}(1149,\cdot)\)
\(\chi_{12935}(1269,\cdot)\)
\(\chi_{12935}(1344,\cdot)\)
\(\chi_{12935}(1399,\cdot)\)
\(\chi_{12935}(1464,\cdot)\)
\(\chi_{12935}(1539,\cdot)\)
\(\chi_{12935}(1669,\cdot)\)
\(\chi_{12935}(1734,\cdot)\)
\(\chi_{12935}(1789,\cdot)\)
\(\chi_{12935}(1864,\cdot)\)
\(\chi_{12935}(2059,\cdot)\)
\(\chi_{12935}(2124,\cdot)\)
\(\chi_{12935}(2179,\cdot)\)
\(\chi_{12935}(2309,\cdot)\)
\(\chi_{12935}(2374,\cdot)\)
\(\chi_{12935}(2384,\cdot)\)
...
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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 };
\((7762,1991,6371)\) → \((-1,i,e\left(\frac{71}{198}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 12935 }(684, a) \) |
\(1\) | \(1\) | \(e\left(\frac{301}{396}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{103}{198}\right)\) | \(e\left(\frac{245}{396}\right)\) | \(e\left(\frac{67}{396}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{92}{99}\right)\) |
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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)
