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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([0,51,80]))
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gp:[g,chi] = znchar(Mod(581, 1152))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1152.581");
| Modulus: | \(1152\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1152\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(96\) |
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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);
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\(\chi_{1152}(5,\cdot)\)
\(\chi_{1152}(29,\cdot)\)
\(\chi_{1152}(77,\cdot)\)
\(\chi_{1152}(101,\cdot)\)
\(\chi_{1152}(149,\cdot)\)
\(\chi_{1152}(173,\cdot)\)
\(\chi_{1152}(221,\cdot)\)
\(\chi_{1152}(245,\cdot)\)
\(\chi_{1152}(293,\cdot)\)
\(\chi_{1152}(317,\cdot)\)
\(\chi_{1152}(365,\cdot)\)
\(\chi_{1152}(389,\cdot)\)
\(\chi_{1152}(437,\cdot)\)
\(\chi_{1152}(461,\cdot)\)
\(\chi_{1152}(509,\cdot)\)
\(\chi_{1152}(533,\cdot)\)
\(\chi_{1152}(581,\cdot)\)
\(\chi_{1152}(605,\cdot)\)
\(\chi_{1152}(653,\cdot)\)
\(\chi_{1152}(677,\cdot)\)
\(\chi_{1152}(725,\cdot)\)
\(\chi_{1152}(749,\cdot)\)
\(\chi_{1152}(797,\cdot)\)
\(\chi_{1152}(821,\cdot)\)
\(\chi_{1152}(869,\cdot)\)
\(\chi_{1152}(893,\cdot)\)
\(\chi_{1152}(941,\cdot)\)
\(\chi_{1152}(965,\cdot)\)
\(\chi_{1152}(1013,\cdot)\)
\(\chi_{1152}(1037,\cdot)\)
...
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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 };
\((127,901,641)\) → \((1,e\left(\frac{17}{32}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1152 }(581, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
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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)
