from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,20,46]))
pari: [g,chi] = znchar(Mod(42,2015))
Basic properties
Modulus: | \(2015\) | |
Conductor: | \(2015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2015.gu
\(\chi_{2015}(42,\cdot)\) \(\chi_{2015}(48,\cdot)\) \(\chi_{2015}(282,\cdot)\) \(\chi_{2015}(393,\cdot)\) \(\chi_{2015}(477,\cdot)\) \(\chi_{2015}(737,\cdot)\) \(\chi_{2015}(757,\cdot)\) \(\chi_{2015}(848,\cdot)\) \(\chi_{2015}(952,\cdot)\) \(\chi_{2015}(1088,\cdot)\) \(\chi_{2015}(1257,\cdot)\) \(\chi_{2015}(1283,\cdot)\) \(\chi_{2015}(1543,\cdot)\) \(\chi_{2015}(1563,\cdot)\) \(\chi_{2015}(1602,\cdot)\) \(\chi_{2015}(1758,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((807,1861,716)\) → \((i,e\left(\frac{1}{3}\right),e\left(\frac{23}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
\( \chi_{ 2015 }(42, a) \) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) |
sage: chi.jacobi_sum(n)