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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256025, base_ring=CyclotomicField(630))
M = H._module
chi = DirichletCharacter(H, M([567,435,189,595]))
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gp:[g,chi] = znchar(Mod(51994, 256025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256025.51994");
| Modulus: | \(256025\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(256025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(630\) |
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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);
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\(\chi_{256025}(1839,\cdot)\)
\(\chi_{256025}(2504,\cdot)\)
\(\chi_{256025}(3594,\cdot)\)
\(\chi_{256025}(3834,\cdot)\)
\(\chi_{256025}(9369,\cdot)\)
\(\chi_{256025}(13219,\cdot)\)
\(\chi_{256025}(13389,\cdot)\)
\(\chi_{256025}(14054,\cdot)\)
\(\chi_{256025}(15384,\cdot)\)
\(\chi_{256025}(15419,\cdot)\)
\(\chi_{256025}(18889,\cdot)\)
\(\chi_{256025}(19554,\cdot)\)
\(\chi_{256025}(20884,\cdot)\)
\(\chi_{256025}(23119,\cdot)\)
\(\chi_{256025}(24664,\cdot)\)
\(\chi_{256025}(25329,\cdot)\)
\(\chi_{256025}(26659,\cdot)\)
\(\chi_{256025}(28514,\cdot)\)
\(\chi_{256025}(29179,\cdot)\)
\(\chi_{256025}(30714,\cdot)\)
\(\chi_{256025}(32709,\cdot)\)
\(\chi_{256025}(34669,\cdot)\)
\(\chi_{256025}(38414,\cdot)\)
\(\chi_{256025}(39079,\cdot)\)
\(\chi_{256025}(40169,\cdot)\)
\(\chi_{256025}(40409,\cdot)\)
\(\chi_{256025}(49794,\cdot)\)
\(\chi_{256025}(49964,\cdot)\)
\(\chi_{256025}(50629,\cdot)\)
\(\chi_{256025}(51994,\cdot)\)
...
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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 };
\((112652,198551,232751,67376)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{29}{42}\right),e\left(\frac{3}{10}\right),e\left(\frac{17}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 256025 }(51994, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{61}{630}\right)\) | \(e\left(\frac{421}{630}\right)\) | \(e\left(\frac{61}{315}\right)\) | \(e\left(\frac{241}{315}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{106}{315}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{286}{315}\right)\) | \(e\left(\frac{122}{315}\right)\) | \(e\left(\frac{67}{630}\right)\) |
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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)
