from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,13]))
pari: [g,chi] = znchar(Mod(451,1225))
Basic properties
Modulus: | \(1225\) | |
Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{49}(10,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1225.bg
\(\chi_{1225}(26,\cdot)\) \(\chi_{1225}(101,\cdot)\) \(\chi_{1225}(201,\cdot)\) \(\chi_{1225}(376,\cdot)\) \(\chi_{1225}(451,\cdot)\) \(\chi_{1225}(551,\cdot)\) \(\chi_{1225}(626,\cdot)\) \(\chi_{1225}(726,\cdot)\) \(\chi_{1225}(801,\cdot)\) \(\chi_{1225}(976,\cdot)\) \(\chi_{1225}(1076,\cdot)\) \(\chi_{1225}(1151,\cdot)\)
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 42 polynomial |
Values on generators
\((1177,101)\) → \((1,e\left(\frac{13}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 1225 }(451, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) |
sage: chi.jacobi_sum(n)