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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9295, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([585,468,560]))
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gp:[g,chi] = znchar(Mod(4618, 9295))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9295.4618");
| Modulus: | \(9295\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9295\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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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);
|
\(\chi_{9295}(3,\cdot)\)
\(\chi_{9295}(42,\cdot)\)
\(\chi_{9295}(48,\cdot)\)
\(\chi_{9295}(113,\cdot)\)
\(\chi_{9295}(152,\cdot)\)
\(\chi_{9295}(302,\cdot)\)
\(\chi_{9295}(328,\cdot)\)
\(\chi_{9295}(367,\cdot)\)
\(\chi_{9295}(412,\cdot)\)
\(\chi_{9295}(432,\cdot)\)
\(\chi_{9295}(438,\cdot)\)
\(\chi_{9295}(477,\cdot)\)
\(\chi_{9295}(542,\cdot)\)
\(\chi_{9295}(588,\cdot)\)
\(\chi_{9295}(718,\cdot)\)
\(\chi_{9295}(757,\cdot)\)
\(\chi_{9295}(763,\cdot)\)
\(\chi_{9295}(828,\cdot)\)
\(\chi_{9295}(1017,\cdot)\)
\(\chi_{9295}(1043,\cdot)\)
\(\chi_{9295}(1082,\cdot)\)
\(\chi_{9295}(1127,\cdot)\)
\(\chi_{9295}(1147,\cdot)\)
\(\chi_{9295}(1153,\cdot)\)
\(\chi_{9295}(1192,\cdot)\)
\(\chi_{9295}(1257,\cdot)\)
\(\chi_{9295}(1303,\cdot)\)
\(\chi_{9295}(1368,\cdot)\)
\(\chi_{9295}(1413,\cdot)\)
\(\chi_{9295}(1433,\cdot)\)
...
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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 };
\((7437,4226,6931)\) → \((-i,e\left(\frac{3}{5}\right),e\left(\frac{28}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 9295 }(4618, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{53}{780}\right)\) | \(e\left(\frac{59}{780}\right)\) | \(e\left(\frac{53}{390}\right)\) | \(e\left(\frac{28}{195}\right)\) | \(e\left(\frac{601}{780}\right)\) | \(e\left(\frac{53}{260}\right)\) | \(e\left(\frac{59}{390}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{53}{195}\right)\) |
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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)
