from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,21,50]))
pari: [g,chi] = znchar(Mod(103,2100))
Basic properties
| Modulus: | \(2100\) | |
| Conductor: | \(700\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{700}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2100.dl
\(\chi_{2100}(103,\cdot)\) \(\chi_{2100}(187,\cdot)\) \(\chi_{2100}(283,\cdot)\) \(\chi_{2100}(367,\cdot)\) \(\chi_{2100}(523,\cdot)\) \(\chi_{2100}(703,\cdot)\) \(\chi_{2100}(787,\cdot)\) \(\chi_{2100}(1027,\cdot)\) \(\chi_{2100}(1123,\cdot)\) \(\chi_{2100}(1363,\cdot)\) \(\chi_{2100}(1447,\cdot)\) \(\chi_{2100}(1627,\cdot)\) \(\chi_{2100}(1783,\cdot)\) \(\chi_{2100}(1867,\cdot)\) \(\chi_{2100}(1963,\cdot)\) \(\chi_{2100}(2047,\cdot)\)
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
\((1051,701,1177,1501)\) → \((-1,1,e\left(\frac{7}{20}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2100 }(103, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)