from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2014, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([4,27]))
pari: [g,chi] = znchar(Mod(23,2014))
Basic properties
Modulus: | \(2014\) | |
Conductor: | \(1007\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1007}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2014.u
\(\chi_{2014}(23,\cdot)\) \(\chi_{2014}(613,\cdot)\) \(\chi_{2014}(719,\cdot)\) \(\chi_{2014}(765,\cdot)\) \(\chi_{2014}(871,\cdot)\) \(\chi_{2014}(1355,\cdot)\) \(\chi_{2014}(1461,\cdot)\) \(\chi_{2014}(1507,\cdot)\) \(\chi_{2014}(1567,\cdot)\) \(\chi_{2014}(1613,\cdot)\) \(\chi_{2014}(1719,\cdot)\) \(\chi_{2014}(1885,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((743,267)\) → \((e\left(\frac{1}{9}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
\( \chi_{ 2014 }(23, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) |
sage: chi.jacobi_sum(n)