from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,21,20]))
pari: [g,chi] = znchar(Mod(1003,2100))
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 \(\chi_{700}(303,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2100.dn
\(\chi_{2100}(67,\cdot)\) \(\chi_{2100}(163,\cdot)\) \(\chi_{2100}(247,\cdot)\) \(\chi_{2100}(403,\cdot)\) \(\chi_{2100}(487,\cdot)\) \(\chi_{2100}(583,\cdot)\) \(\chi_{2100}(667,\cdot)\) \(\chi_{2100}(823,\cdot)\) \(\chi_{2100}(1003,\cdot)\) \(\chi_{2100}(1087,\cdot)\) \(\chi_{2100}(1327,\cdot)\) \(\chi_{2100}(1423,\cdot)\) \(\chi_{2100}(1663,\cdot)\) \(\chi_{2100}(1747,\cdot)\) \(\chi_{2100}(1927,\cdot)\) \(\chi_{2100}(2083,\cdot)\)
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
\((1051,701,1177,1501)\) → \((-1,1,e\left(\frac{7}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2100 }(1003, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)