from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(94, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([42]))
pari: [g,chi] = znchar(Mod(37,94))
Basic properties
Modulus: | \(94\) | |
Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{47}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 94.c
\(\chi_{94}(3,\cdot)\) \(\chi_{94}(7,\cdot)\) \(\chi_{94}(9,\cdot)\) \(\chi_{94}(17,\cdot)\) \(\chi_{94}(21,\cdot)\) \(\chi_{94}(25,\cdot)\) \(\chi_{94}(27,\cdot)\) \(\chi_{94}(37,\cdot)\) \(\chi_{94}(49,\cdot)\) \(\chi_{94}(51,\cdot)\) \(\chi_{94}(53,\cdot)\) \(\chi_{94}(55,\cdot)\) \(\chi_{94}(59,\cdot)\) \(\chi_{94}(61,\cdot)\) \(\chi_{94}(63,\cdot)\) \(\chi_{94}(65,\cdot)\) \(\chi_{94}(71,\cdot)\) \(\chi_{94}(75,\cdot)\) \(\chi_{94}(79,\cdot)\) \(\chi_{94}(81,\cdot)\) \(\chi_{94}(83,\cdot)\) \(\chi_{94}(89,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\(5\) → \(e\left(\frac{21}{23}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 94 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) |
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)