from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,3,5]))
pari: [g,chi] = znchar(Mod(1879,2100))
Basic properties
Modulus: | \(2100\) | |
Conductor: | \(700\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{700}(479,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2100.cy
\(\chi_{2100}(19,\cdot)\) \(\chi_{2100}(439,\cdot)\) \(\chi_{2100}(619,\cdot)\) \(\chi_{2100}(859,\cdot)\) \(\chi_{2100}(1039,\cdot)\) \(\chi_{2100}(1279,\cdot)\) \(\chi_{2100}(1459,\cdot)\) \(\chi_{2100}(1879,\cdot)\)
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: | 30.30.639471349555952501681804656982421875000000000000000000000000000000.1 |
Values on generators
\((1051,701,1177,1501)\) → \((-1,1,e\left(\frac{1}{10}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2100 }(1879, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)