from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(430, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,8]))
pari: [g,chi] = znchar(Mod(111,430))
Basic properties
Modulus: | \(430\) | |
Conductor: | \(43\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{43}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 430.q
\(\chi_{430}(31,\cdot)\) \(\chi_{430}(81,\cdot)\) \(\chi_{430}(101,\cdot)\) \(\chi_{430}(111,\cdot)\) \(\chi_{430}(181,\cdot)\) \(\chi_{430}(271,\cdot)\) \(\chi_{430}(281,\cdot)\) \(\chi_{430}(311,\cdot)\) \(\chi_{430}(341,\cdot)\) \(\chi_{430}(361,\cdot)\) \(\chi_{430}(401,\cdot)\) \(\chi_{430}(411,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((87,261)\) → \((1,e\left(\frac{4}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 430 }(111, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) |
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)