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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5625, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([250,111]))
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gp:[g,chi] = znchar(Mod(257, 5625))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5625.257");
| Modulus: | \(5625\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1125\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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 \(\chi_{1125}(347,\cdot)\) |
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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: | no |
| 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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\(\chi_{5625}(32,\cdot)\)
\(\chi_{5625}(218,\cdot)\)
\(\chi_{5625}(257,\cdot)\)
\(\chi_{5625}(293,\cdot)\)
\(\chi_{5625}(407,\cdot)\)
\(\chi_{5625}(482,\cdot)\)
\(\chi_{5625}(518,\cdot)\)
\(\chi_{5625}(632,\cdot)\)
\(\chi_{5625}(668,\cdot)\)
\(\chi_{5625}(707,\cdot)\)
\(\chi_{5625}(743,\cdot)\)
\(\chi_{5625}(857,\cdot)\)
\(\chi_{5625}(893,\cdot)\)
\(\chi_{5625}(968,\cdot)\)
\(\chi_{5625}(1082,\cdot)\)
\(\chi_{5625}(1118,\cdot)\)
\(\chi_{5625}(1157,\cdot)\)
\(\chi_{5625}(1343,\cdot)\)
\(\chi_{5625}(1382,\cdot)\)
\(\chi_{5625}(1418,\cdot)\)
\(\chi_{5625}(1532,\cdot)\)
\(\chi_{5625}(1607,\cdot)\)
\(\chi_{5625}(1643,\cdot)\)
\(\chi_{5625}(1757,\cdot)\)
\(\chi_{5625}(1793,\cdot)\)
\(\chi_{5625}(1832,\cdot)\)
\(\chi_{5625}(1868,\cdot)\)
\(\chi_{5625}(1982,\cdot)\)
\(\chi_{5625}(2018,\cdot)\)
\(\chi_{5625}(2093,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{300})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 300 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((4376,1252)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{37}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5625 }(257, a) \) |
\(1\) | \(1\) | \(e\left(\frac{61}{300}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{29}{300}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{33}{50}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
