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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(379045, base_ring=CyclotomicField(3010))
M = H._module
chi = DirichletCharacter(H, M([1505,301,835]))
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gp:[g,chi] = znchar(Mod(6544, 379045))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("379045.6544");
| Modulus: | \(379045\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(379045\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(3010\) |
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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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\(\chi_{379045}(414,\cdot)\)
\(\chi_{379045}(1384,\cdot)\)
\(\chi_{379045}(1704,\cdot)\)
\(\chi_{379045}(3264,\cdot)\)
\(\chi_{379045}(3639,\cdot)\)
\(\chi_{379045}(3694,\cdot)\)
\(\chi_{379045}(4289,\cdot)\)
\(\chi_{379045}(4494,\cdot)\)
\(\chi_{379045}(4719,\cdot)\)
\(\chi_{379045}(4904,\cdot)\)
\(\chi_{379045}(4924,\cdot)\)
\(\chi_{379045}(4984,\cdot)\)
\(\chi_{379045}(5334,\cdot)\)
\(\chi_{379045}(6009,\cdot)\)
\(\chi_{379045}(6214,\cdot)\)
\(\chi_{379045}(6544,\cdot)\)
\(\chi_{379045}(6624,\cdot)\)
\(\chi_{379045}(6919,\cdot)\)
\(\chi_{379045}(6974,\cdot)\)
\(\chi_{379045}(7944,\cdot)\)
\(\chi_{379045}(8149,\cdot)\)
\(\chi_{379045}(8264,\cdot)\)
\(\chi_{379045}(8559,\cdot)\)
\(\chi_{379045}(8799,\cdot)\)
\(\chi_{379045}(9229,\cdot)\)
\(\chi_{379045}(10199,\cdot)\)
\(\chi_{379045}(10519,\cdot)\)
\(\chi_{379045}(12079,\cdot)\)
\(\chi_{379045}(12454,\cdot)\)
\(\chi_{379045}(12509,\cdot)\)
...
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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 };
\((303237,286596,188601)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{167}{602}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 379045 }(6544, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{48}{1505}\right)\) | \(e\left(\frac{167}{602}\right)\) | \(e\left(\frac{96}{1505}\right)\) | \(e\left(\frac{133}{430}\right)\) | \(e\left(\frac{157}{430}\right)\) | \(e\left(\frac{144}{1505}\right)\) | \(e\left(\frac{167}{301}\right)\) | \(e\left(\frac{2293}{3010}\right)\) | \(e\left(\frac{1027}{3010}\right)\) | \(e\left(\frac{158}{1505}\right)\) |
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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)
