from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,51,50]))
pari: [g,chi] = znchar(Mod(47,1400))
Basic properties
Modulus: | \(1400\) | |
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 \(\chi_{700}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1400.dl
\(\chi_{1400}(47,\cdot)\) \(\chi_{1400}(87,\cdot)\) \(\chi_{1400}(103,\cdot)\) \(\chi_{1400}(327,\cdot)\) \(\chi_{1400}(367,\cdot)\) \(\chi_{1400}(383,\cdot)\) \(\chi_{1400}(423,\cdot)\) \(\chi_{1400}(647,\cdot)\) \(\chi_{1400}(663,\cdot)\) \(\chi_{1400}(703,\cdot)\) \(\chi_{1400}(887,\cdot)\) \(\chi_{1400}(927,\cdot)\) \(\chi_{1400}(983,\cdot)\) \(\chi_{1400}(1167,\cdot)\) \(\chi_{1400}(1223,\cdot)\) \(\chi_{1400}(1263,\cdot)\)
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
\((351,701,1177,801)\) → \((-1,1,e\left(\frac{17}{20}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 1400 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)