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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31734, base_ring=CyclotomicField(840))
M = H._module
chi = DirichletCharacter(H, M([420,189,260]))
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gp:[g,chi] = znchar(Mod(2807, 31734))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31734.2807");
| 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: | \(5289\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(840\) |
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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 \(\chi_{5289}(2807,\cdot)\) |
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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: | 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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\(\chi_{31734}(71,\cdot)\)
\(\chi_{31734}(89,\cdot)\)
\(\chi_{31734}(233,\cdot)\)
\(\chi_{31734}(485,\cdot)\)
\(\chi_{31734}(503,\cdot)\)
\(\chi_{31734}(521,\cdot)\)
\(\chi_{31734}(593,\cdot)\)
\(\chi_{31734}(1223,\cdot)\)
\(\chi_{31734}(1259,\cdot)\)
\(\chi_{31734}(1277,\cdot)\)
\(\chi_{31734}(1295,\cdot)\)
\(\chi_{31734}(1961,\cdot)\)
\(\chi_{31734}(1997,\cdot)\)
\(\chi_{31734}(2033,\cdot)\)
\(\chi_{31734}(2069,\cdot)\)
\(\chi_{31734}(2393,\cdot)\)
\(\chi_{31734}(2555,\cdot)\)
\(\chi_{31734}(2609,\cdot)\)
\(\chi_{31734}(2699,\cdot)\)
\(\chi_{31734}(2735,\cdot)\)
\(\chi_{31734}(2771,\cdot)\)
\(\chi_{31734}(2807,\cdot)\)
\(\chi_{31734}(2987,\cdot)\)
\(\chi_{31734}(3185,\cdot)\)
\(\chi_{31734}(3473,\cdot)\)
\(\chi_{31734}(3509,\cdot)\)
\(\chi_{31734}(3545,\cdot)\)
\(\chi_{31734}(3761,\cdot)\)
\(\chi_{31734}(4247,\cdot)\)
\(\chi_{31734}(4283,\cdot)\)
...
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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: |
$\Q(\zeta_{840})$ |
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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 840 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((14105,30961,22879)\) → \((-1,e\left(\frac{9}{40}\right),e\left(\frac{13}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 31734 }(2807, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{129}{280}\right)\) | \(e\left(\frac{739}{840}\right)\) | \(e\left(\frac{577}{840}\right)\) | \(e\left(\frac{761}{840}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{643}{840}\right)\) | \(e\left(\frac{173}{210}\right)\) |
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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)
