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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2646, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([98,33]))
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gp:[g,chi] = znchar(Mod(1237, 2646))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2646.1237");
| Modulus: | \(2646\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1323\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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 \(\chi_{1323}(1237,\cdot)\) |
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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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\(\chi_{2646}(103,\cdot)\)
\(\chi_{2646}(115,\cdot)\)
\(\chi_{2646}(229,\cdot)\)
\(\chi_{2646}(241,\cdot)\)
\(\chi_{2646}(355,\cdot)\)
\(\chi_{2646}(367,\cdot)\)
\(\chi_{2646}(481,\cdot)\)
\(\chi_{2646}(493,\cdot)\)
\(\chi_{2646}(733,\cdot)\)
\(\chi_{2646}(745,\cdot)\)
\(\chi_{2646}(859,\cdot)\)
\(\chi_{2646}(871,\cdot)\)
\(\chi_{2646}(985,\cdot)\)
\(\chi_{2646}(997,\cdot)\)
\(\chi_{2646}(1111,\cdot)\)
\(\chi_{2646}(1123,\cdot)\)
\(\chi_{2646}(1237,\cdot)\)
\(\chi_{2646}(1249,\cdot)\)
\(\chi_{2646}(1363,\cdot)\)
\(\chi_{2646}(1375,\cdot)\)
\(\chi_{2646}(1615,\cdot)\)
\(\chi_{2646}(1627,\cdot)\)
\(\chi_{2646}(1741,\cdot)\)
\(\chi_{2646}(1753,\cdot)\)
\(\chi_{2646}(1867,\cdot)\)
\(\chi_{2646}(1879,\cdot)\)
\(\chi_{2646}(1993,\cdot)\)
\(\chi_{2646}(2005,\cdot)\)
\(\chi_{2646}(2119,\cdot)\)
\(\chi_{2646}(2131,\cdot)\)
...
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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: |
$\Q(\zeta_{63})$ |
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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 126 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((785,1081)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{11}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2646 }(1237, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(-1\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{21}\right)\) |
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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)
