from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2450, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,26]))
pari: [g,chi] = znchar(Mod(149,2450))
Basic properties
Modulus: | \(2450\) | |
Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{245}(149,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2450.be
\(\chi_{2450}(149,\cdot)\) \(\chi_{2450}(249,\cdot)\) \(\chi_{2450}(499,\cdot)\) \(\chi_{2450}(599,\cdot)\) \(\chi_{2450}(849,\cdot)\) \(\chi_{2450}(1199,\cdot)\) \(\chi_{2450}(1299,\cdot)\) \(\chi_{2450}(1649,\cdot)\) \(\chi_{2450}(1899,\cdot)\) \(\chi_{2450}(1999,\cdot)\) \(\chi_{2450}(2249,\cdot)\) \(\chi_{2450}(2349,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | 42.42.8050468075656610214837511220114705524038488445061950919170859146595001220703125.1 |
Values on generators
\((1177,101)\) → \((-1,e\left(\frac{13}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 2450 }(149, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)