from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(495, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,15,42]))
pari: [g,chi] = znchar(Mod(7,495))
Basic properties
| Modulus: | \(495\) | |
| Conductor: | \(495\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 495.bs
\(\chi_{495}(7,\cdot)\) \(\chi_{495}(13,\cdot)\) \(\chi_{495}(52,\cdot)\) \(\chi_{495}(112,\cdot)\) \(\chi_{495}(178,\cdot)\) \(\chi_{495}(193,\cdot)\) \(\chi_{495}(238,\cdot)\) \(\chi_{495}(277,\cdot)\) \(\chi_{495}(283,\cdot)\) \(\chi_{495}(292,\cdot)\) \(\chi_{495}(337,\cdot)\) \(\chi_{495}(358,\cdot)\) \(\chi_{495}(382,\cdot)\) \(\chi_{495}(403,\cdot)\) \(\chi_{495}(448,\cdot)\) \(\chi_{495}(457,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((56,397,46)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 495 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)