from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4275, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,16]))
pari: [g,chi] = znchar(Mod(332,4275))
Basic properties
Modulus: | \(4275\) | |
Conductor: | \(285\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{285}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4275.ff
\(\chi_{4275}(332,\cdot)\) \(\chi_{4275}(557,\cdot)\) \(\chi_{4275}(593,\cdot)\) \(\chi_{4275}(1043,\cdot)\) \(\chi_{4275}(1232,\cdot)\) \(\chi_{4275}(1943,\cdot)\) \(\chi_{4275}(2132,\cdot)\) \(\chi_{4275}(2582,\cdot)\) \(\chi_{4275}(3068,\cdot)\) \(\chi_{4275}(3293,\cdot)\) \(\chi_{4275}(3482,\cdot)\) \(\chi_{4275}(3968,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1901,1027,1351)\) → \((-1,i,e\left(\frac{4}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 4275 }(332, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) |
sage: chi.jacobi_sum(n)