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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58080, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,0,43]))
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gp:[g,chi] = znchar(Mod(5761, 58080))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58080.5761");
| Modulus: | \(58080\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(121\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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 \(\chi_{121}(74,\cdot)\) |
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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);
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\(\chi_{58080}(1921,\cdot)\)
\(\chi_{58080}(3841,\cdot)\)
\(\chi_{58080}(5761,\cdot)\)
\(\chi_{58080}(7201,\cdot)\)
\(\chi_{58080}(8641,\cdot)\)
\(\chi_{58080}(9121,\cdot)\)
\(\chi_{58080}(11041,\cdot)\)
\(\chi_{58080}(12481,\cdot)\)
\(\chi_{58080}(13921,\cdot)\)
\(\chi_{58080}(14401,\cdot)\)
\(\chi_{58080}(16321,\cdot)\)
\(\chi_{58080}(17761,\cdot)\)
\(\chi_{58080}(19201,\cdot)\)
\(\chi_{58080}(19681,\cdot)\)
\(\chi_{58080}(21601,\cdot)\)
\(\chi_{58080}(23041,\cdot)\)
\(\chi_{58080}(24481,\cdot)\)
\(\chi_{58080}(24961,\cdot)\)
\(\chi_{58080}(26881,\cdot)\)
\(\chi_{58080}(28321,\cdot)\)
\(\chi_{58080}(29761,\cdot)\)
\(\chi_{58080}(32161,\cdot)\)
\(\chi_{58080}(33601,\cdot)\)
\(\chi_{58080}(35041,\cdot)\)
\(\chi_{58080}(35521,\cdot)\)
\(\chi_{58080}(37441,\cdot)\)
\(\chi_{58080}(40321,\cdot)\)
\(\chi_{58080}(40801,\cdot)\)
\(\chi_{58080}(42721,\cdot)\)
\(\chi_{58080}(44161,\cdot)\)
...
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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 };
\((32671,50821,19361,11617,14401)\) → \((1,1,1,1,e\left(\frac{43}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 58080 }(5761, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{17}{22}\right)\) |
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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)
