from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7056, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,7,5]))
pari: [g,chi] = znchar(Mod(47,7056))
Basic properties
Modulus: | \(7056\) | |
Conductor: | \(1764\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1764}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7056.ft
\(\chi_{7056}(47,\cdot)\) \(\chi_{7056}(1055,\cdot)\) \(\chi_{7056}(1823,\cdot)\) \(\chi_{7056}(2063,\cdot)\) \(\chi_{7056}(2831,\cdot)\) \(\chi_{7056}(3071,\cdot)\) \(\chi_{7056}(3839,\cdot)\) \(\chi_{7056}(4079,\cdot)\) \(\chi_{7056}(4847,\cdot)\) \(\chi_{7056}(5087,\cdot)\) \(\chi_{7056}(5855,\cdot)\) \(\chi_{7056}(6863,\cdot)\)
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: | 42.0.594905448297071958649415344409202411272524582191201645168397283612406235485559112254331131690310420404371456.1 |
Values on generators
\((6175,1765,785,4609)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{5}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 7056 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{21}\right)\) |
sage: chi.jacobi_sum(n)