from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1075, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,26]))
pari: [g,chi] = znchar(Mod(101,1075))
Basic properties
Modulus: | \(1075\) | |
Conductor: | \(43\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{43}(15,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1075.x
\(\chi_{1075}(101,\cdot)\) \(\chi_{1075}(126,\cdot)\) \(\chi_{1075}(326,\cdot)\) \(\chi_{1075}(401,\cdot)\) \(\chi_{1075}(526,\cdot)\) \(\chi_{1075}(576,\cdot)\) \(\chi_{1075}(626,\cdot)\) \(\chi_{1075}(676,\cdot)\) \(\chi_{1075}(701,\cdot)\) \(\chi_{1075}(726,\cdot)\) \(\chi_{1075}(826,\cdot)\) \(\chi_{1075}(926,\cdot)\)
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
\((302,476)\) → \((1,e\left(\frac{13}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 1075 }(101, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) |
sage: chi.jacobi_sum(n)