from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4650, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,57,16]))
pari: [g,chi] = znchar(Mod(113,4650))
Basic properties
| Modulus: | \(4650\) | |
| Conductor: | \(2325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2325}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4650.gk
\(\chi_{4650}(113,\cdot)\) \(\chi_{4650}(617,\cdot)\) \(\chi_{4650}(917,\cdot)\) \(\chi_{4650}(1223,\cdot)\) \(\chi_{4650}(1373,\cdot)\) \(\chi_{4650}(1733,\cdot)\) \(\chi_{4650}(1787,\cdot)\) \(\chi_{4650}(2003,\cdot)\) \(\chi_{4650}(2033,\cdot)\) \(\chi_{4650}(2363,\cdot)\) \(\chi_{4650}(2747,\cdot)\) \(\chi_{4650}(3203,\cdot)\) \(\chi_{4650}(3827,\cdot)\) \(\chi_{4650}(3947,\cdot)\) \(\chi_{4650}(3977,\cdot)\) \(\chi_{4650}(4037,\cdot)\)
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
\((3101,2977,1801)\) → \((-1,e\left(\frac{19}{20}\right),e\left(\frac{4}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4650 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) |
sage: chi.jacobi_sum(n)