from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6240, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,21,0,0,2]))
pari: [g,chi] = znchar(Mod(691,6240))
Basic properties
| Modulus: | \(6240\) | |
| Conductor: | \(416\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{416}(275,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6240.pk
\(\chi_{6240}(691,\cdot)\) \(\chi_{6240}(1051,\cdot)\) \(\chi_{6240}(1891,\cdot)\) \(\chi_{6240}(2971,\cdot)\) \(\chi_{6240}(3811,\cdot)\) \(\chi_{6240}(4171,\cdot)\) \(\chi_{6240}(5011,\cdot)\) \(\chi_{6240}(6091,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.24.31808511574029960248322509834333516654369310400053248.2 |
Values on generators
\((1951,2341,2081,2497,5761)\) → \((-1,e\left(\frac{7}{8}\right),1,1,e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6240 }(691, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(i\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{24}\right)\) |
sage: chi.jacobi_sum(n)