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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2607, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([65,13,60]))
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gp:[g,chi] = znchar(Mod(1223, 2607))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2607.1223");
| Modulus: | \(2607\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2607\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(130\) |
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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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\(\chi_{2607}(8,\cdot)\)
\(\chi_{2607}(62,\cdot)\)
\(\chi_{2607}(101,\cdot)\)
\(\chi_{2607}(299,\cdot)\)
\(\chi_{2607}(326,\cdot)\)
\(\chi_{2607}(338,\cdot)\)
\(\chi_{2607}(380,\cdot)\)
\(\chi_{2607}(413,\cdot)\)
\(\chi_{2607}(512,\cdot)\)
\(\chi_{2607}(536,\cdot)\)
\(\chi_{2607}(563,\cdot)\)
\(\chi_{2607}(776,\cdot)\)
\(\chi_{2607}(800,\cdot)\)
\(\chi_{2607}(842,\cdot)\)
\(\chi_{2607}(854,\cdot)\)
\(\chi_{2607}(887,\cdot)\)
\(\chi_{2607}(986,\cdot)\)
\(\chi_{2607}(1073,\cdot)\)
\(\chi_{2607}(1091,\cdot)\)
\(\chi_{2607}(1124,\cdot)\)
\(\chi_{2607}(1223,\cdot)\)
\(\chi_{2607}(1250,\cdot)\)
\(\chi_{2607}(1316,\cdot)\)
\(\chi_{2607}(1328,\cdot)\)
\(\chi_{2607}(1361,\cdot)\)
\(\chi_{2607}(1460,\cdot)\)
\(\chi_{2607}(1487,\cdot)\)
\(\chi_{2607}(1547,\cdot)\)
\(\chi_{2607}(1553,\cdot)\)
\(\chi_{2607}(1568,\cdot)\)
...
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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 };
\((1739,475,793)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{6}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 2607 }(1223, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) |
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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)
