from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4950, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([10,15,12]))
pari: [g,chi] = znchar(Mod(49,4950))
Basic properties
Modulus: | \(4950\) | |
Conductor: | \(495\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{495}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4950.el
\(\chi_{4950}(49,\cdot)\) \(\chi_{4950}(499,\cdot)\) \(\chi_{4950}(949,\cdot)\) \(\chi_{4950}(1699,\cdot)\) \(\chi_{4950}(2149,\cdot)\) \(\chi_{4950}(2599,\cdot)\) \(\chi_{4950}(3199,\cdot)\) \(\chi_{4950}(4849,\cdot)\)
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: | Number field defined by a degree 30 polynomial |
Values on generators
\((551,2377,4501)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{2}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 4950 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)