from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([9,20]))
pari: [g,chi] = znchar(Mod(39,350))
Basic properties
Modulus: | \(350\) | |
Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{175}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 350.u
\(\chi_{350}(9,\cdot)\) \(\chi_{350}(39,\cdot)\) \(\chi_{350}(79,\cdot)\) \(\chi_{350}(109,\cdot)\) \(\chi_{350}(179,\cdot)\) \(\chi_{350}(219,\cdot)\) \(\chi_{350}(289,\cdot)\) \(\chi_{350}(319,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.30.35434884492252294752034913472016341984272003173828125.1 |
Values on generators
\((127,101)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 350 }(39, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) |
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)