from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,2,7]))
pari: [g,chi] = znchar(Mod(1577,1911))
Basic properties
Modulus: | \(1911\) | |
Conductor: | \(1911\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1911.ds
\(\chi_{1911}(179,\cdot)\) \(\chi_{1911}(212,\cdot)\) \(\chi_{1911}(452,\cdot)\) \(\chi_{1911}(485,\cdot)\) \(\chi_{1911}(725,\cdot)\) \(\chi_{1911}(758,\cdot)\) \(\chi_{1911}(1031,\cdot)\) \(\chi_{1911}(1271,\cdot)\) \(\chi_{1911}(1544,\cdot)\) \(\chi_{1911}(1577,\cdot)\) \(\chi_{1911}(1817,\cdot)\) \(\chi_{1911}(1850,\cdot)\)
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: | 42.0.171796661872303139426286971553175729564906991908179786155620948893904232254867038476961967376575716647979404317071.1 |
Values on generators
\((638,1522,1471)\) → \((-1,e\left(\frac{1}{21}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 1911 }(1577, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(-1\) | \(e\left(\frac{4}{21}\right)\) |
sage: chi.jacobi_sum(n)