from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,20,21,10]))
pari: [g,chi] = znchar(Mod(709,6300))
Basic properties
Modulus: | \(6300\) | |
Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1575}(709,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6300.ft
\(\chi_{6300}(709,\cdot)\) \(\chi_{6300}(1969,\cdot)\) \(\chi_{6300}(2209,\cdot)\) \(\chi_{6300}(3229,\cdot)\) \(\chi_{6300}(3469,\cdot)\) \(\chi_{6300}(4489,\cdot)\) \(\chi_{6300}(4729,\cdot)\) \(\chi_{6300}(5989,\cdot)\)
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
\((3151,2801,3277,3601)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 6300 }(709, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)