from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,40]))
pari: [g,chi] = znchar(Mod(501,1225))
Basic properties
| Modulus: | \(1225\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1225.x
\(\chi_{1225}(51,\cdot)\) \(\chi_{1225}(151,\cdot)\) \(\chi_{1225}(326,\cdot)\) \(\chi_{1225}(401,\cdot)\) \(\chi_{1225}(501,\cdot)\) \(\chi_{1225}(576,\cdot)\) \(\chi_{1225}(676,\cdot)\) \(\chi_{1225}(751,\cdot)\) \(\chi_{1225}(926,\cdot)\) \(\chi_{1225}(1026,\cdot)\) \(\chi_{1225}(1101,\cdot)\) \(\chi_{1225}(1201,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((1177,101)\) → \((1,e\left(\frac{20}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(501, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) |
sage: chi.jacobi_sum(n)