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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11200, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,45,24,0]))
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gp:[g,chi] = znchar(Mod(7589, 11200))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11200.7589");
| Modulus: | \(11200\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1600\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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 \(\chi_{1600}(1189,\cdot)\) |
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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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\(\chi_{11200}(29,\cdot)\)
\(\chi_{11200}(309,\cdot)\)
\(\chi_{11200}(589,\cdot)\)
\(\chi_{11200}(869,\cdot)\)
\(\chi_{11200}(1429,\cdot)\)
\(\chi_{11200}(1709,\cdot)\)
\(\chi_{11200}(1989,\cdot)\)
\(\chi_{11200}(2269,\cdot)\)
\(\chi_{11200}(2829,\cdot)\)
\(\chi_{11200}(3109,\cdot)\)
\(\chi_{11200}(3389,\cdot)\)
\(\chi_{11200}(3669,\cdot)\)
\(\chi_{11200}(4229,\cdot)\)
\(\chi_{11200}(4509,\cdot)\)
\(\chi_{11200}(4789,\cdot)\)
\(\chi_{11200}(5069,\cdot)\)
\(\chi_{11200}(5629,\cdot)\)
\(\chi_{11200}(5909,\cdot)\)
\(\chi_{11200}(6189,\cdot)\)
\(\chi_{11200}(6469,\cdot)\)
\(\chi_{11200}(7029,\cdot)\)
\(\chi_{11200}(7309,\cdot)\)
\(\chi_{11200}(7589,\cdot)\)
\(\chi_{11200}(7869,\cdot)\)
\(\chi_{11200}(8429,\cdot)\)
\(\chi_{11200}(8709,\cdot)\)
\(\chi_{11200}(8989,\cdot)\)
\(\chi_{11200}(9269,\cdot)\)
\(\chi_{11200}(9829,\cdot)\)
\(\chi_{11200}(10109,\cdot)\)
...
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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 };
\((5951,10501,5377,6401)\) → \((1,e\left(\frac{9}{16}\right),e\left(\frac{3}{10}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 11200 }(7589, a) \) |
\(1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
