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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24003, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([21,0,106]))
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gp:[g,chi] = znchar(Mod(9458, 24003))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("24003.9458");
| Modulus: | \(24003\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1143\) |
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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 \(\chi_{1143}(695,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{24003}(197,\cdot)\)
\(\chi_{24003}(1898,\cdot)\)
\(\chi_{24003}(3473,\cdot)\)
\(\chi_{24003}(3851,\cdot)\)
\(\chi_{24003}(5111,\cdot)\)
\(\chi_{24003}(5930,\cdot)\)
\(\chi_{24003}(6686,\cdot)\)
\(\chi_{24003}(6812,\cdot)\)
\(\chi_{24003}(6875,\cdot)\)
\(\chi_{24003}(7379,\cdot)\)
\(\chi_{24003}(7631,\cdot)\)
\(\chi_{24003}(9458,\cdot)\)
\(\chi_{24003}(10781,\cdot)\)
\(\chi_{24003}(10844,\cdot)\)
\(\chi_{24003}(11915,\cdot)\)
\(\chi_{24003}(12734,\cdot)\)
\(\chi_{24003}(13238,\cdot)\)
\(\chi_{24003}(14435,\cdot)\)
\(\chi_{24003}(15128,\cdot)\)
\(\chi_{24003}(15947,\cdot)\)
\(\chi_{24003}(16073,\cdot)\)
\(\chi_{24003}(16325,\cdot)\)
\(\chi_{24003}(17207,\cdot)\)
\(\chi_{24003}(17396,\cdot)\)
\(\chi_{24003}(17774,\cdot)\)
\(\chi_{24003}(18530,\cdot)\)
\(\chi_{24003}(19727,\cdot)\)
\(\chi_{24003}(20483,\cdot)\)
\(\chi_{24003}(20609,\cdot)\)
\(\chi_{24003}(20672,\cdot)\)
...
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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 };
\((11558,17146,7750)\) → \((e\left(\frac{1}{6}\right),1,e\left(\frac{53}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 24003 }(9458, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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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)
