from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(679, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,27]))
pari: [g,chi] = znchar(Mod(601,679))
Basic properties
| Modulus: | \(679\) | |
| Conductor: | \(679\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | 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 679.bq
\(\chi_{679}(20,\cdot)\) \(\chi_{679}(34,\cdot)\) \(\chi_{679}(55,\cdot)\) \(\chi_{679}(69,\cdot)\) \(\chi_{679}(125,\cdot)\) \(\chi_{679}(139,\cdot)\) \(\chi_{679}(160,\cdot)\) \(\chi_{679}(174,\cdot)\) \(\chi_{679}(272,\cdot)\) \(\chi_{679}(321,\cdot)\) \(\chi_{679}(342,\cdot)\) \(\chi_{679}(433,\cdot)\) \(\chi_{679}(440,\cdot)\) \(\chi_{679}(531,\cdot)\) \(\chi_{679}(552,\cdot)\) \(\chi_{679}(601,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.32.1292684086480957274390235903780365469718835315185422412744961998012838206753.1 |
Values on generators
\((486,393)\) → \((-1,e\left(\frac{27}{32}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 679 }(601, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
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)