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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(292215, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([330,165,440,384,360]))
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gp:[g,chi] = znchar(Mod(41717, 292215))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("292215.41717");
| Modulus: | \(292215\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(292215\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(660\) |
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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);
|
\(\chi_{292215}(422,\cdot)\)
\(\chi_{292215}(863,\cdot)\)
\(\chi_{292215}(6218,\cdot)\)
\(\chi_{292215}(10322,\cdot)\)
\(\chi_{292215}(10448,\cdot)\)
\(\chi_{292215}(12413,\cdot)\)
\(\chi_{292215}(17147,\cdot)\)
\(\chi_{292215}(17378,\cdot)\)
\(\chi_{292215}(18797,\cdot)\)
\(\chi_{292215}(19793,\cdot)\)
\(\chi_{292215}(21767,\cdot)\)
\(\chi_{292215}(23048,\cdot)\)
\(\chi_{292215}(23972,\cdot)\)
\(\chi_{292215}(25358,\cdot)\)
\(\chi_{292215}(25463,\cdot)\)
\(\chi_{292215}(25727,\cdot)\)
\(\chi_{292215}(28142,\cdot)\)
\(\chi_{292215}(31397,\cdot)\)
\(\chi_{292215}(31838,\cdot)\)
\(\chi_{292215}(32708,\cdot)\)
\(\chi_{292215}(33707,\cdot)\)
\(\chi_{292215}(33812,\cdot)\)
\(\chi_{292215}(36668,\cdot)\)
\(\chi_{292215}(41057,\cdot)\)
\(\chi_{292215}(41528,\cdot)\)
\(\chi_{292215}(41717,\cdot)\)
\(\chi_{292215}(42683,\cdot)\)
\(\chi_{292215}(44762,\cdot)\)
\(\chi_{292215}(48878,\cdot)\)
\(\chi_{292215}(49877,\cdot)\)
...
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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 };
\((97406,116887,166981,57961,38116)\) → \((-1,i,e\left(\frac{2}{3}\right),e\left(\frac{32}{55}\right),e\left(\frac{6}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(26\) | \(29\) | \(31\) |
| \( \chi_{ 292215 }(41717, a) \) |
\(1\) | \(1\) | \(e\left(\frac{499}{660}\right)\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{491}{660}\right)\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{161}{165}\right)\) |
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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)
