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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31734, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([80,117,100]))
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gp:[g,chi] = znchar(Mod(7, 31734))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31734.7");
| Modulus: | \(31734\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(15867\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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 \(\chi_{15867}(7,\cdot)\) |
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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);
|
\(\chi_{31734}(7,\cdot)\)
\(\chi_{31734}(1327,\cdot)\)
\(\chi_{31734}(3103,\cdot)\)
\(\chi_{31734}(5425,\cdot)\)
\(\chi_{31734}(5971,\cdot)\)
\(\chi_{31734}(8293,\cdot)\)
\(\chi_{31734}(8521,\cdot)\)
\(\chi_{31734}(9067,\cdot)\)
\(\chi_{31734}(9295,\cdot)\)
\(\chi_{31734}(10069,\cdot)\)
\(\chi_{31734}(10843,\cdot)\)
\(\chi_{31734}(12937,\cdot)\)
\(\chi_{31734}(13711,\cdot)\)
\(\chi_{31734}(14485,\cdot)\)
\(\chi_{31734}(14713,\cdot)\)
\(\chi_{31734}(15259,\cdot)\)
\(\chi_{31734}(15487,\cdot)\)
\(\chi_{31734}(17809,\cdot)\)
\(\chi_{31734}(18355,\cdot)\)
\(\chi_{31734}(20677,\cdot)\)
\(\chi_{31734}(22453,\cdot)\)
\(\chi_{31734}(23773,\cdot)\)
\(\chi_{31734}(24547,\cdot)\)
\(\chi_{31734}(24775,\cdot)\)
\(\chi_{31734}(25321,\cdot)\)
\(\chi_{31734}(25549,\cdot)\)
\(\chi_{31734}(26095,\cdot)\)
\(\chi_{31734}(29419,\cdot)\)
\(\chi_{31734}(29965,\cdot)\)
\(\chi_{31734}(30193,\cdot)\)
...
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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: |
$\Q(\zeta_{120})$ |
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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 120 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((14105,30961,22879)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{39}{40}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 31734 }(7, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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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)
