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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(257725, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([78,45,65]))
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gp:[g,chi] = znchar(Mod(7779, 257725))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("257725.7779");
| Modulus: | \(257725\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(257725\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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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: | 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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\(\chi_{257725}(814,\cdot)\)
\(\chi_{257725}(3814,\cdot)\)
\(\chi_{257725}(3944,\cdot)\)
\(\chi_{257725}(4779,\cdot)\)
\(\chi_{257725}(4909,\cdot)\)
\(\chi_{257725}(7779,\cdot)\)
\(\chi_{257725}(7909,\cdot)\)
\(\chi_{257725}(8744,\cdot)\)
\(\chi_{257725}(11744,\cdot)\)
\(\chi_{257725}(12709,\cdot)\)
\(\chi_{257725}(12839,\cdot)\)
\(\chi_{257725}(15709,\cdot)\)
\(\chi_{257725}(15839,\cdot)\)
\(\chi_{257725}(16804,\cdot)\)
\(\chi_{257725}(19804,\cdot)\)
\(\chi_{257725}(20639,\cdot)\)
\(\chi_{257725}(20769,\cdot)\)
\(\chi_{257725}(23639,\cdot)\)
\(\chi_{257725}(23769,\cdot)\)
\(\chi_{257725}(24734,\cdot)\)
\(\chi_{257725}(27604,\cdot)\)
\(\chi_{257725}(27734,\cdot)\)
\(\chi_{257725}(28569,\cdot)\)
\(\chi_{257725}(31569,\cdot)\)
\(\chi_{257725}(32534,\cdot)\)
\(\chi_{257725}(32664,\cdot)\)
\(\chi_{257725}(35534,\cdot)\)
\(\chi_{257725}(35664,\cdot)\)
\(\chi_{257725}(36629,\cdot)\)
\(\chi_{257725}(39629,\cdot)\)
...
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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 };
\((144327,193676,177451)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{3}{52}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 257725 }(7779, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{195}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{94}{195}\right)\) | \(e\left(\frac{116}{195}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{163}{195}\right)\) | \(e\left(\frac{389}{390}\right)\) |
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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)
