from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4275, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([10,3,0]))
pari: [g,chi] = znchar(Mod(229,4275))
Basic properties
Modulus: | \(4275\) | |
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 \(\chi_{225}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4275.ei
\(\chi_{4275}(229,\cdot)\) \(\chi_{4275}(1084,\cdot)\) \(\chi_{4275}(1654,\cdot)\) \(\chi_{4275}(1939,\cdot)\) \(\chi_{4275}(2509,\cdot)\) \(\chi_{4275}(2794,\cdot)\) \(\chi_{4275}(3364,\cdot)\) \(\chi_{4275}(4219,\cdot)\)
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
\((1901,1027,1351)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 4275 }(229, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{30}\right)\) |
sage: chi.jacobi_sum(n)