from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,15,20,18]))
pari: [g,chi] = znchar(Mod(137,4800))
Basic properties
Modulus: | \(4800\) | |
Conductor: | \(2400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2400}(2237,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4800.ew
\(\chi_{4800}(137,\cdot)\) \(\chi_{4800}(473,\cdot)\) \(\chi_{4800}(617,\cdot)\) \(\chi_{4800}(953,\cdot)\) \(\chi_{4800}(1097,\cdot)\) \(\chi_{4800}(1433,\cdot)\) \(\chi_{4800}(1577,\cdot)\) \(\chi_{4800}(1913,\cdot)\) \(\chi_{4800}(2537,\cdot)\) \(\chi_{4800}(2873,\cdot)\) \(\chi_{4800}(3017,\cdot)\) \(\chi_{4800}(3353,\cdot)\) \(\chi_{4800}(3497,\cdot)\) \(\chi_{4800}(3833,\cdot)\) \(\chi_{4800}(3977,\cdot)\) \(\chi_{4800}(4313,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | 40.40.1348884380735497228084799435251384320000000000000000000000000000000000000000000000000000000000000000000000.2 |
Values on generators
\((4351,901,1601,577)\) → \((1,e\left(\frac{3}{8}\right),-1,e\left(\frac{9}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 4800 }(137, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)