from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4928, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,20,6]))
pari: [g,chi] = znchar(Mod(79,4928))
Basic properties
Modulus: | \(4928\) | |
Conductor: | \(1232\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1232}(1003,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4928.fc
\(\chi_{4928}(79,\cdot)\) \(\chi_{4928}(303,\cdot)\) \(\chi_{4928}(431,\cdot)\) \(\chi_{4928}(655,\cdot)\) \(\chi_{4928}(975,\cdot)\) \(\chi_{4928}(1327,\cdot)\) \(\chi_{4928}(1647,\cdot)\) \(\chi_{4928}(1999,\cdot)\) \(\chi_{4928}(2543,\cdot)\) \(\chi_{4928}(2767,\cdot)\) \(\chi_{4928}(2895,\cdot)\) \(\chi_{4928}(3119,\cdot)\) \(\chi_{4928}(3439,\cdot)\) \(\chi_{4928}(3791,\cdot)\) \(\chi_{4928}(4111,\cdot)\) \(\chi_{4928}(4463,\cdot)\)
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
\((4159,1541,2817,3137)\) → \((-1,i,e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 4928 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) |
sage: chi.jacobi_sum(n)