from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1300, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,21,20]))
pari: [g,chi] = znchar(Mod(3,1300))
Basic properties
| Modulus: | \(1300\) | |
| Conductor: | \(1300\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1300.cu
\(\chi_{1300}(3,\cdot)\) \(\chi_{1300}(87,\cdot)\) \(\chi_{1300}(263,\cdot)\) \(\chi_{1300}(347,\cdot)\) \(\chi_{1300}(367,\cdot)\) \(\chi_{1300}(503,\cdot)\) \(\chi_{1300}(523,\cdot)\) \(\chi_{1300}(627,\cdot)\) \(\chi_{1300}(763,\cdot)\) \(\chi_{1300}(783,\cdot)\) \(\chi_{1300}(867,\cdot)\) \(\chi_{1300}(887,\cdot)\) \(\chi_{1300}(1023,\cdot)\) \(\chi_{1300}(1127,\cdot)\) \(\chi_{1300}(1147,\cdot)\) \(\chi_{1300}(1283,\cdot)\)
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
\((651,677,301)\) → \((-1,e\left(\frac{7}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1300 }(3, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) |
sage: chi.jacobi_sum(n)