from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,33,20]))
pari: [g,chi] = znchar(Mod(23,1400))
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 \(\chi_{700}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1400.dn
\(\chi_{1400}(23,\cdot)\) \(\chi_{1400}(247,\cdot)\) \(\chi_{1400}(263,\cdot)\) \(\chi_{1400}(303,\cdot)\) \(\chi_{1400}(487,\cdot)\) \(\chi_{1400}(527,\cdot)\) \(\chi_{1400}(583,\cdot)\) \(\chi_{1400}(767,\cdot)\) \(\chi_{1400}(823,\cdot)\) \(\chi_{1400}(863,\cdot)\) \(\chi_{1400}(1047,\cdot)\) \(\chi_{1400}(1087,\cdot)\) \(\chi_{1400}(1103,\cdot)\) \(\chi_{1400}(1327,\cdot)\) \(\chi_{1400}(1367,\cdot)\) \(\chi_{1400}(1383,\cdot)\)
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
\((351,701,1177,801)\) → \((-1,1,e\left(\frac{11}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1400 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) |
sage: chi.jacobi_sum(n)