from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5400, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,20,9]))
pari: [g,chi] = znchar(Mod(289,5400))
Basic properties
Modulus: | \(5400\) | |
Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{225}(214,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5400.dx
\(\chi_{5400}(289,\cdot)\) \(\chi_{5400}(1009,\cdot)\) \(\chi_{5400}(1369,\cdot)\) \(\chi_{5400}(2089,\cdot)\) \(\chi_{5400}(3169,\cdot)\) \(\chi_{5400}(3529,\cdot)\) \(\chi_{5400}(4609,\cdot)\) \(\chi_{5400}(5329,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.30.5399088047333990303844331037907977588474750518798828125.1 |
Values on generators
\((1351,2701,1001,2377)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 5400 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)