from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3900, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,54,25]))
pari: [g,chi] = znchar(Mod(19,3900))
Basic properties
Modulus: | \(3900\) | |
Conductor: | \(1300\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1300}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3900.ft
\(\chi_{3900}(19,\cdot)\) \(\chi_{3900}(319,\cdot)\) \(\chi_{3900}(379,\cdot)\) \(\chi_{3900}(739,\cdot)\) \(\chi_{3900}(1159,\cdot)\) \(\chi_{3900}(1519,\cdot)\) \(\chi_{3900}(1579,\cdot)\) \(\chi_{3900}(1879,\cdot)\) \(\chi_{3900}(1939,\cdot)\) \(\chi_{3900}(2359,\cdot)\) \(\chi_{3900}(2659,\cdot)\) \(\chi_{3900}(2719,\cdot)\) \(\chi_{3900}(3079,\cdot)\) \(\chi_{3900}(3139,\cdot)\) \(\chi_{3900}(3439,\cdot)\) \(\chi_{3900}(3859,\cdot)\)
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
\((1951,1301,3277,301)\) → \((-1,1,e\left(\frac{9}{10}\right),e\left(\frac{5}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3900 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)