from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,15,8]))
pari: [g,chi] = znchar(Mod(454,6045))
Basic properties
| Modulus: | \(6045\) | |
| Conductor: | \(2015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2015}(454,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6045.jo
\(\chi_{6045}(454,\cdot)\) \(\chi_{6045}(844,\cdot)\) \(\chi_{6045}(3574,\cdot)\) \(\chi_{6045}(3769,\cdot)\) \(\chi_{6045}(4354,\cdot)\) \(\chi_{6045}(4939,\cdot)\) \(\chi_{6045}(5134,\cdot)\) \(\chi_{6045}(5329,\cdot)\)
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: | Number field defined by a degree 30 polynomial |
Values on generators
\((4031,4837,1861,2731)\) → \((1,-1,-1,e\left(\frac{4}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 6045 }(454, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) |
sage: chi.jacobi_sum(n)