from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,21,5]))
pari: [g,chi] = znchar(Mod(1109,2100))
Basic properties
| Modulus: | \(2100\) | |
| Conductor: | \(525\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{525}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2100.cv
\(\chi_{2100}(89,\cdot)\) \(\chi_{2100}(269,\cdot)\) \(\chi_{2100}(509,\cdot)\) \(\chi_{2100}(689,\cdot)\) \(\chi_{2100}(929,\cdot)\) \(\chi_{2100}(1109,\cdot)\) \(\chi_{2100}(1529,\cdot)\) \(\chi_{2100}(1769,\cdot)\)
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
\((1051,701,1177,1501)\) → \((1,-1,e\left(\frac{7}{10}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2100 }(1109, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)