from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,3,40]))
pari: [g,chi] = znchar(Mod(277,1400))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1400.dm
\(\chi_{1400}(37,\cdot)\) \(\chi_{1400}(53,\cdot)\) \(\chi_{1400}(277,\cdot)\) \(\chi_{1400}(317,\cdot)\) \(\chi_{1400}(333,\cdot)\) \(\chi_{1400}(373,\cdot)\) \(\chi_{1400}(597,\cdot)\) \(\chi_{1400}(613,\cdot)\) \(\chi_{1400}(653,\cdot)\) \(\chi_{1400}(837,\cdot)\) \(\chi_{1400}(877,\cdot)\) \(\chi_{1400}(933,\cdot)\) \(\chi_{1400}(1117,\cdot)\) \(\chi_{1400}(1173,\cdot)\) \(\chi_{1400}(1213,\cdot)\) \(\chi_{1400}(1397,\cdot)\)
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
\((351,701,1177,801)\) → \((1,-1,e\left(\frac{1}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1400 }(277, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)