from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,40]))
pari: [g,chi] = znchar(Mod(109,1764))
Basic properties
Modulus: | \(1764\) | |
Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{49}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1764.bx
\(\chi_{1764}(37,\cdot)\) \(\chi_{1764}(109,\cdot)\) \(\chi_{1764}(289,\cdot)\) \(\chi_{1764}(541,\cdot)\) \(\chi_{1764}(613,\cdot)\) \(\chi_{1764}(793,\cdot)\) \(\chi_{1764}(865,\cdot)\) \(\chi_{1764}(1045,\cdot)\) \(\chi_{1764}(1117,\cdot)\) \(\chi_{1764}(1297,\cdot)\) \(\chi_{1764}(1369,\cdot)\) \(\chi_{1764}(1621,\cdot)\)
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 21 polynomial |
Values on generators
\((883,785,1081)\) → \((1,1,e\left(\frac{20}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 1764 }(109, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{10}{21}\right)\) |
sage: chi.jacobi_sum(n)