from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4650, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,6,29]))
pari: [g,chi] = znchar(Mod(641,4650))
Basic properties
| Modulus: | \(4650\) | |
| Conductor: | \(2325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2325}(641,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4650.fe
\(\chi_{4650}(641,\cdot)\) \(\chi_{4650}(1481,\cdot)\) \(\chi_{4650}(1811,\cdot)\) \(\chi_{4650}(1841,\cdot)\) \(\chi_{4650}(2111,\cdot)\) \(\chi_{4650}(2471,\cdot)\) \(\chi_{4650}(2621,\cdot)\) \(\chi_{4650}(3731,\cdot)\)
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: | 30.30.905457666226479689370802194763207293493326652338016202747894567437469959259033203125.1 |
Values on generators
\((3101,2977,1801)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{29}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4650 }(641, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{30}\right)\) |
sage: chi.jacobi_sum(n)