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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(152269, base_ring=CyclotomicField(312))
M = H._module
chi = DirichletCharacter(H, M([236,117,174]))
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gp:[g,chi] = znchar(Mod(29884, 152269))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("152269.29884");
| Modulus: | \(152269\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(152269\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(312\) |
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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: | yes |
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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);
|
\(\chi_{152269}(875,\cdot)\)
\(\chi_{152269}(2831,\cdot)\)
\(\chi_{152269}(3715,\cdot)\)
\(\chi_{152269}(6146,\cdot)\)
\(\chi_{152269}(8005,\cdot)\)
\(\chi_{152269}(11970,\cdot)\)
\(\chi_{152269}(14941,\cdot)\)
\(\chi_{152269}(15649,\cdot)\)
\(\chi_{152269}(19049,\cdot)\)
\(\chi_{152269}(19406,\cdot)\)
\(\chi_{152269}(20154,\cdot)\)
\(\chi_{152269}(29884,\cdot)\)
\(\chi_{152269}(31373,\cdot)\)
\(\chi_{152269}(36423,\cdot)\)
\(\chi_{152269}(39374,\cdot)\)
\(\chi_{152269}(39550,\cdot)\)
\(\chi_{152269}(39771,\cdot)\)
\(\chi_{152269}(41981,\cdot)\)
\(\chi_{152269}(44600,\cdot)\)
\(\chi_{152269}(45959,\cdot)\)
\(\chi_{152269}(46310,\cdot)\)
\(\chi_{152269}(47506,\cdot)\)
\(\chi_{152269}(47772,\cdot)\)
\(\chi_{152269}(52504,\cdot)\)
\(\chi_{152269}(53161,\cdot)\)
\(\chi_{152269}(57288,\cdot)\)
\(\chi_{152269}(58387,\cdot)\)
\(\chi_{152269}(58692,\cdot)\)
\(\chi_{152269}(61695,\cdot)\)
\(\chi_{152269}(63385,\cdot)\)
...
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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 };
\((149567,143313,83318)\) → \((e\left(\frac{59}{78}\right),e\left(\frac{3}{8}\right),e\left(\frac{29}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 152269 }(29884, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{203}{312}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{93}{104}\right)\) | \(e\left(\frac{67}{312}\right)\) | \(e\left(\frac{271}{312}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{275}{312}\right)\) |
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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)
