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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(79184, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,0,115,63]))
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gp:[g,chi] = znchar(Mod(42033, 79184))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("79184.42033");
| Modulus: | \(79184\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4949\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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 \(\chi_{4949}(2441,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{79184}(17,\cdot)\)
\(\chi_{79184}(3953,\cdot)\)
\(\chi_{79184}(6529,\cdot)\)
\(\chi_{79184}(7985,\cdot)\)
\(\chi_{79184}(8097,\cdot)\)
\(\chi_{79184}(9761,\cdot)\)
\(\chi_{79184}(11217,\cdot)\)
\(\chi_{79184}(11329,\cdot)\)
\(\chi_{79184}(15265,\cdot)\)
\(\chi_{79184}(17841,\cdot)\)
\(\chi_{79184}(18497,\cdot)\)
\(\chi_{79184}(19297,\cdot)\)
\(\chi_{79184}(19409,\cdot)\)
\(\chi_{79184}(21073,\cdot)\)
\(\chi_{79184}(22529,\cdot)\)
\(\chi_{79184}(22641,\cdot)\)
\(\chi_{79184}(29153,\cdot)\)
\(\chi_{79184}(29809,\cdot)\)
\(\chi_{79184}(30609,\cdot)\)
\(\chi_{79184}(30721,\cdot)\)
\(\chi_{79184}(32385,\cdot)\)
\(\chi_{79184}(33953,\cdot)\)
\(\chi_{79184}(37889,\cdot)\)
\(\chi_{79184}(40465,\cdot)\)
\(\chi_{79184}(41121,\cdot)\)
\(\chi_{79184}(41921,\cdot)\)
\(\chi_{79184}(42033,\cdot)\)
\(\chi_{79184}(43697,\cdot)\)
\(\chi_{79184}(45153,\cdot)\)
\(\chi_{79184}(45265,\cdot)\)
...
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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 };
\((29695,19797,75953,16465)\) → \((1,1,e\left(\frac{23}{42}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 79184 }(42033, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{17}{105}\right)\) |
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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)
