from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,57,50]))
pari: [g,chi] = znchar(Mod(1013,1400))
Basic properties
Modulus: | \(1400\) | |
Conductor: | \(1400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1400.dk
\(\chi_{1400}(117,\cdot)\) \(\chi_{1400}(173,\cdot)\) \(\chi_{1400}(213,\cdot)\) \(\chi_{1400}(397,\cdot)\) \(\chi_{1400}(437,\cdot)\) \(\chi_{1400}(453,\cdot)\) \(\chi_{1400}(677,\cdot)\) \(\chi_{1400}(717,\cdot)\) \(\chi_{1400}(733,\cdot)\) \(\chi_{1400}(773,\cdot)\) \(\chi_{1400}(997,\cdot)\) \(\chi_{1400}(1013,\cdot)\) \(\chi_{1400}(1053,\cdot)\) \(\chi_{1400}(1237,\cdot)\) \(\chi_{1400}(1277,\cdot)\) \(\chi_{1400}(1333,\cdot)\)
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
\((351,701,1177,801)\) → \((1,-1,e\left(\frac{19}{20}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 1400 }(1013, a) \) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) |
sage: chi.jacobi_sum(n)