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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2416, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([75,75,104]))
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gp:[g,chi] = znchar(Mod(871, 2416))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2416.871");
| Modulus: | \(2416\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1208\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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 \(\chi_{1208}(267,\cdot)\) |
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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: | no |
| 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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\(\chi_{2416}(39,\cdot)\)
\(\chi_{2416}(55,\cdot)\)
\(\chi_{2416}(103,\cdot)\)
\(\chi_{2416}(231,\cdot)\)
\(\chi_{2416}(295,\cdot)\)
\(\chi_{2416}(327,\cdot)\)
\(\chi_{2416}(423,\cdot)\)
\(\chi_{2416}(439,\cdot)\)
\(\chi_{2416}(471,\cdot)\)
\(\chi_{2416}(487,\cdot)\)
\(\chi_{2416}(615,\cdot)\)
\(\chi_{2416}(647,\cdot)\)
\(\chi_{2416}(743,\cdot)\)
\(\chi_{2416}(791,\cdot)\)
\(\chi_{2416}(855,\cdot)\)
\(\chi_{2416}(871,\cdot)\)
\(\chi_{2416}(951,\cdot)\)
\(\chi_{2416}(1079,\cdot)\)
\(\chi_{2416}(1239,\cdot)\)
\(\chi_{2416}(1255,\cdot)\)
\(\chi_{2416}(1303,\cdot)\)
\(\chi_{2416}(1399,\cdot)\)
\(\chi_{2416}(1447,\cdot)\)
\(\chi_{2416}(1495,\cdot)\)
\(\chi_{2416}(1527,\cdot)\)
\(\chi_{2416}(1559,\cdot)\)
\(\chi_{2416}(1607,\cdot)\)
\(\chi_{2416}(1655,\cdot)\)
\(\chi_{2416}(1671,\cdot)\)
\(\chi_{2416}(1703,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{75})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 150 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((303,1813,1969)\) → \((-1,-1,e\left(\frac{52}{75}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2416 }(871, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{139}{150}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{150}\right)\) |
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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)
