from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1296, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,11]))
pari: [g,chi] = znchar(Mod(23,1296))
Basic properties
Modulus: | \(1296\) | |
Conductor: | \(648\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{648}(347,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1296.bn
\(\chi_{1296}(23,\cdot)\) \(\chi_{1296}(119,\cdot)\) \(\chi_{1296}(167,\cdot)\) \(\chi_{1296}(263,\cdot)\) \(\chi_{1296}(311,\cdot)\) \(\chi_{1296}(407,\cdot)\) \(\chi_{1296}(455,\cdot)\) \(\chi_{1296}(551,\cdot)\) \(\chi_{1296}(599,\cdot)\) \(\chi_{1296}(695,\cdot)\) \(\chi_{1296}(743,\cdot)\) \(\chi_{1296}(839,\cdot)\) \(\chi_{1296}(887,\cdot)\) \(\chi_{1296}(983,\cdot)\) \(\chi_{1296}(1031,\cdot)\) \(\chi_{1296}(1127,\cdot)\) \(\chi_{1296}(1175,\cdot)\) \(\chi_{1296}(1271,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((1135,325,1217)\) → \((-1,-1,e\left(\frac{11}{54}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 1296 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) |
sage: chi.jacobi_sum(n)