from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,21,0,16]))
pari: [g,chi] = znchar(Mod(361,2240))
Basic properties
| Modulus: | \(2240\) | |
| Conductor: | \(224\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{224}(109,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2240.ev
\(\chi_{2240}(121,\cdot)\) \(\chi_{2240}(361,\cdot)\) \(\chi_{2240}(681,\cdot)\) \(\chi_{2240}(921,\cdot)\) \(\chi_{2240}(1241,\cdot)\) \(\chi_{2240}(1481,\cdot)\) \(\chi_{2240}(1801,\cdot)\) \(\chi_{2240}(2041,\cdot)\)
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: | 24.24.329123002999201416128761938882499016916992.1 |
Values on generators
\((1471,1541,897,1921)\) → \((1,e\left(\frac{7}{8}\right),1,e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 2240 }(361, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)