from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1296, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,0,32]))
pari: [g,chi] = znchar(Mod(49,1296))
Basic properties
Modulus: | \(1296\) | |
Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{81}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1296.bg
\(\chi_{1296}(49,\cdot)\) \(\chi_{1296}(97,\cdot)\) \(\chi_{1296}(193,\cdot)\) \(\chi_{1296}(241,\cdot)\) \(\chi_{1296}(337,\cdot)\) \(\chi_{1296}(385,\cdot)\) \(\chi_{1296}(481,\cdot)\) \(\chi_{1296}(529,\cdot)\) \(\chi_{1296}(625,\cdot)\) \(\chi_{1296}(673,\cdot)\) \(\chi_{1296}(769,\cdot)\) \(\chi_{1296}(817,\cdot)\) \(\chi_{1296}(913,\cdot)\) \(\chi_{1296}(961,\cdot)\) \(\chi_{1296}(1057,\cdot)\) \(\chi_{1296}(1105,\cdot)\) \(\chi_{1296}(1201,\cdot)\) \(\chi_{1296}(1249,\cdot)\)
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 27 polynomial |
Values on generators
\((1135,325,1217)\) → \((1,1,e\left(\frac{16}{27}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 1296 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) |
sage: chi.jacobi_sum(n)