from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,0,2]))
pari: [g,chi] = znchar(Mod(313,2015))
Basic properties
| Modulus: | \(2015\) | |
| Conductor: | \(155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{155}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2015.ha
\(\chi_{2015}(53,\cdot)\) \(\chi_{2015}(313,\cdot)\) \(\chi_{2015}(352,\cdot)\) \(\chi_{2015}(482,\cdot)\) \(\chi_{2015}(508,\cdot)\) \(\chi_{2015}(703,\cdot)\) \(\chi_{2015}(768,\cdot)\) \(\chi_{2015}(1067,\cdot)\) \(\chi_{2015}(1158,\cdot)\) \(\chi_{2015}(1262,\cdot)\) \(\chi_{2015}(1288,\cdot)\) \(\chi_{2015}(1522,\cdot)\) \(\chi_{2015}(1717,\cdot)\) \(\chi_{2015}(1873,\cdot)\) \(\chi_{2015}(1912,\cdot)\) \(\chi_{2015}(1977,\cdot)\)
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
\((807,1861,716)\) → \((-i,1,e\left(\frac{1}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 2015 }(313, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) |
sage: chi.jacobi_sum(n)