from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3100, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,39,56]))
pari: [g,chi] = znchar(Mod(317,3100))
Basic properties
Modulus: | \(3100\) | |
Conductor: | \(775\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{775}(317,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3100.fz
\(\chi_{3100}(173,\cdot)\) \(\chi_{3100}(317,\cdot)\) \(\chi_{3100}(413,\cdot)\) \(\chi_{3100}(537,\cdot)\) \(\chi_{3100}(733,\cdot)\) \(\chi_{3100}(753,\cdot)\) \(\chi_{3100}(1433,\cdot)\) \(\chi_{3100}(1497,\cdot)\) \(\chi_{3100}(2153,\cdot)\) \(\chi_{3100}(2313,\cdot)\) \(\chi_{3100}(2437,\cdot)\) \(\chi_{3100}(2477,\cdot)\) \(\chi_{3100}(2717,\cdot)\) \(\chi_{3100}(2777,\cdot)\) \(\chi_{3100}(2897,\cdot)\) \(\chi_{3100}(2973,\cdot)\)
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
\((1551,2977,1801)\) → \((1,e\left(\frac{13}{20}\right),e\left(\frac{14}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 3100 }(317, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)