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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7225, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([68,217]))
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gp:[g,chi] = znchar(Mod(1782, 7225))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7225.1782");
| Modulus: | \(7225\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1445\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(272\) |
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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: | no, induced from \(\chi_{1445}(337,\cdot)\) |
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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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\(\chi_{7225}(82,\cdot)\)
\(\chi_{7225}(107,\cdot)\)
\(\chi_{7225}(143,\cdot)\)
\(\chi_{7225}(182,\cdot)\)
\(\chi_{7225}(193,\cdot)\)
\(\chi_{7225}(207,\cdot)\)
\(\chi_{7225}(368,\cdot)\)
\(\chi_{7225}(418,\cdot)\)
\(\chi_{7225}(507,\cdot)\)
\(\chi_{7225}(532,\cdot)\)
\(\chi_{7225}(568,\cdot)\)
\(\chi_{7225}(607,\cdot)\)
\(\chi_{7225}(632,\cdot)\)
\(\chi_{7225}(793,\cdot)\)
\(\chi_{7225}(843,\cdot)\)
\(\chi_{7225}(957,\cdot)\)
\(\chi_{7225}(993,\cdot)\)
\(\chi_{7225}(1032,\cdot)\)
\(\chi_{7225}(1043,\cdot)\)
\(\chi_{7225}(1057,\cdot)\)
\(\chi_{7225}(1218,\cdot)\)
\(\chi_{7225}(1268,\cdot)\)
\(\chi_{7225}(1357,\cdot)\)
\(\chi_{7225}(1382,\cdot)\)
\(\chi_{7225}(1418,\cdot)\)
\(\chi_{7225}(1457,\cdot)\)
\(\chi_{7225}(1468,\cdot)\)
\(\chi_{7225}(1482,\cdot)\)
\(\chi_{7225}(1643,\cdot)\)
\(\chi_{7225}(1693,\cdot)\)
...
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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 };
\((2602,2026)\) → \((i,e\left(\frac{217}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 7225 }(1782, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{149}{272}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{103}{272}\right)\) | \(e\left(\frac{247}{272}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{95}{272}\right)\) | \(e\left(\frac{57}{272}\right)\) | \(e\left(\frac{2}{17}\right)\) |
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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)
