from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1960, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,21,34]))
pari: [g,chi] = znchar(Mod(39,1960))
Basic properties
Modulus: | \(1960\) | |
Conductor: | \(980\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{980}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1960.dg
\(\chi_{1960}(39,\cdot)\) \(\chi_{1960}(319,\cdot)\) \(\chi_{1960}(359,\cdot)\) \(\chi_{1960}(599,\cdot)\) \(\chi_{1960}(639,\cdot)\) \(\chi_{1960}(879,\cdot)\) \(\chi_{1960}(919,\cdot)\) \(\chi_{1960}(1159,\cdot)\) \(\chi_{1960}(1199,\cdot)\) \(\chi_{1960}(1479,\cdot)\) \(\chi_{1960}(1719,\cdot)\) \(\chi_{1960}(1759,\cdot)\)
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: | Number field defined by a degree 42 polynomial |
Values on generators
\((1471,981,1177,1081)\) → \((-1,1,-1,e\left(\frac{17}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 1960 }(39, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)