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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,81,72,104]))
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gp:[g,chi] = znchar(Mod(1637, 3648))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3648.1637");
| Modulus: | \(3648\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3648\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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;
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| 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);
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| 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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\(\chi_{3648}(29,\cdot)\)
\(\chi_{3648}(53,\cdot)\)
\(\chi_{3648}(173,\cdot)\)
\(\chi_{3648}(269,\cdot)\)
\(\chi_{3648}(317,\cdot)\)
\(\chi_{3648}(413,\cdot)\)
\(\chi_{3648}(485,\cdot)\)
\(\chi_{3648}(509,\cdot)\)
\(\chi_{3648}(629,\cdot)\)
\(\chi_{3648}(725,\cdot)\)
\(\chi_{3648}(773,\cdot)\)
\(\chi_{3648}(869,\cdot)\)
\(\chi_{3648}(941,\cdot)\)
\(\chi_{3648}(965,\cdot)\)
\(\chi_{3648}(1085,\cdot)\)
\(\chi_{3648}(1181,\cdot)\)
\(\chi_{3648}(1229,\cdot)\)
\(\chi_{3648}(1325,\cdot)\)
\(\chi_{3648}(1397,\cdot)\)
\(\chi_{3648}(1421,\cdot)\)
\(\chi_{3648}(1541,\cdot)\)
\(\chi_{3648}(1637,\cdot)\)
\(\chi_{3648}(1685,\cdot)\)
\(\chi_{3648}(1781,\cdot)\)
\(\chi_{3648}(1853,\cdot)\)
\(\chi_{3648}(1877,\cdot)\)
\(\chi_{3648}(1997,\cdot)\)
\(\chi_{3648}(2093,\cdot)\)
\(\chi_{3648}(2141,\cdot)\)
\(\chi_{3648}(2237,\cdot)\)
...
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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: |
$\Q(\zeta_{144})$ |
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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));
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| Fixed field: |
Number field defined by a degree 144 polynomial (not computed) |
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sage:chi.fixed_field()
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\((2623,2053,1217,1921)\) → \((1,e\left(\frac{9}{16}\right),-1,e\left(\frac{13}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3648 }(1637, a) \) |
\(1\) | \(1\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{83}{144}\right)\) |
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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)
