from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,11]))
pari: [g,chi] = znchar(Mod(188,2013))
Basic properties
Modulus: | \(2013\) | |
Conductor: | \(183\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{183}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2013.ew
\(\chi_{2013}(188,\cdot)\) \(\chi_{2013}(716,\cdot)\) \(\chi_{2013}(980,\cdot)\) \(\chi_{2013}(1178,\cdot)\) \(\chi_{2013}(1442,\cdot)\) \(\chi_{2013}(1574,\cdot)\) \(\chi_{2013}(1805,\cdot)\) \(\chi_{2013}(1937,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.0.85385482345314369605512628350861103547186790293449369086287.1 |
Values on generators
\((1343,1465,1222)\) → \((-1,1,e\left(\frac{11}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 2013 }(188, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)