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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4170, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([69,0,55]))
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gp:[g,chi] = znchar(Mod(641, 4170))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4170.641");
| Modulus: | \(4170\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(417\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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 \(\chi_{417}(224,\cdot)\) |
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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);
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\(\chi_{4170}(101,\cdot)\)
\(\chi_{4170}(161,\cdot)\)
\(\chi_{4170}(281,\cdot)\)
\(\chi_{4170}(371,\cdot)\)
\(\chi_{4170}(401,\cdot)\)
\(\chi_{4170}(521,\cdot)\)
\(\chi_{4170}(551,\cdot)\)
\(\chi_{4170}(641,\cdot)\)
\(\chi_{4170}(671,\cdot)\)
\(\chi_{4170}(821,\cdot)\)
\(\chi_{4170}(851,\cdot)\)
\(\chi_{4170}(1031,\cdot)\)
\(\chi_{4170}(1061,\cdot)\)
\(\chi_{4170}(1301,\cdot)\)
\(\chi_{4170}(1361,\cdot)\)
\(\chi_{4170}(1451,\cdot)\)
\(\chi_{4170}(1541,\cdot)\)
\(\chi_{4170}(1601,\cdot)\)
\(\chi_{4170}(1631,\cdot)\)
\(\chi_{4170}(1661,\cdot)\)
\(\chi_{4170}(1721,\cdot)\)
\(\chi_{4170}(1961,\cdot)\)
\(\chi_{4170}(2081,\cdot)\)
\(\chi_{4170}(2111,\cdot)\)
\(\chi_{4170}(2141,\cdot)\)
\(\chi_{4170}(2381,\cdot)\)
\(\chi_{4170}(2471,\cdot)\)
\(\chi_{4170}(2621,\cdot)\)
\(\chi_{4170}(2681,\cdot)\)
\(\chi_{4170}(2711,\cdot)\)
...
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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 };
\((1391,3337,1531)\) → \((-1,1,e\left(\frac{55}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4170 }(641, a) \) |
\(1\) | \(1\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{109}{138}\right)\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{43}{138}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{133}{138}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{61}{69}\right)\) | \(e\left(\frac{35}{138}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
