from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4650, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,44]))
pari: [g,chi] = znchar(Mod(107,4650))
Basic properties
| Modulus: | \(4650\) | |
| Conductor: | \(465\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{465}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4650.fs
\(\chi_{4650}(107,\cdot)\) \(\chi_{4650}(143,\cdot)\) \(\chi_{4650}(257,\cdot)\) \(\chi_{4650}(293,\cdot)\) \(\chi_{4650}(443,\cdot)\) \(\chi_{4650}(857,\cdot)\) \(\chi_{4650}(1043,\cdot)\) \(\chi_{4650}(1157,\cdot)\) \(\chi_{4650}(1343,\cdot)\) \(\chi_{4650}(3107,\cdot)\) \(\chi_{4650}(3293,\cdot)\) \(\chi_{4650}(3407,\cdot)\) \(\chi_{4650}(3593,\cdot)\) \(\chi_{4650}(3707,\cdot)\) \(\chi_{4650}(3893,\cdot)\) \(\chi_{4650}(4607,\cdot)\)
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
\((3101,2977,1801)\) → \((-1,i,e\left(\frac{11}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4650 }(107, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) |
sage: chi.jacobi_sum(n)