from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1175, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,28]))
pari: [g,chi] = znchar(Mod(24,1175))
Basic properties
Modulus: | \(1175\) | |
Conductor: | \(235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(46\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{235}(24,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1175.o
\(\chi_{1175}(24,\cdot)\) \(\chi_{1175}(49,\cdot)\) \(\chi_{1175}(74,\cdot)\) \(\chi_{1175}(149,\cdot)\) \(\chi_{1175}(224,\cdot)\) \(\chi_{1175}(249,\cdot)\) \(\chi_{1175}(299,\cdot)\) \(\chi_{1175}(324,\cdot)\) \(\chi_{1175}(474,\cdot)\) \(\chi_{1175}(524,\cdot)\) \(\chi_{1175}(549,\cdot)\) \(\chi_{1175}(674,\cdot)\) \(\chi_{1175}(824,\cdot)\) \(\chi_{1175}(849,\cdot)\) \(\chi_{1175}(874,\cdot)\) \(\chi_{1175}(899,\cdot)\) \(\chi_{1175}(949,\cdot)\) \(\chi_{1175}(974,\cdot)\) \(\chi_{1175}(999,\cdot)\) \(\chi_{1175}(1024,\cdot)\) \(\chi_{1175}(1099,\cdot)\) \(\chi_{1175}(1149,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
Fixed field: | 46.46.445262221645814097378614331350194306504897709623466822770698795048558574688434600830078125.1 |
Values on generators
\((377,851)\) → \((-1,e\left(\frac{14}{23}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 1175 }(24, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{9}{46}\right)\) |
sage: chi.jacobi_sum(n)