from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,15,24,32]))
pari: [g,chi] = znchar(Mod(11,3648))
Basic properties
Modulus: | \(3648\) | |
Conductor: | \(3648\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3648.dx
\(\chi_{3648}(11,\cdot)\) \(\chi_{3648}(83,\cdot)\) \(\chi_{3648}(467,\cdot)\) \(\chi_{3648}(539,\cdot)\) \(\chi_{3648}(923,\cdot)\) \(\chi_{3648}(995,\cdot)\) \(\chi_{3648}(1379,\cdot)\) \(\chi_{3648}(1451,\cdot)\) \(\chi_{3648}(1835,\cdot)\) \(\chi_{3648}(1907,\cdot)\) \(\chi_{3648}(2291,\cdot)\) \(\chi_{3648}(2363,\cdot)\) \(\chi_{3648}(2747,\cdot)\) \(\chi_{3648}(2819,\cdot)\) \(\chi_{3648}(3203,\cdot)\) \(\chi_{3648}(3275,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((2623,2053,1217,1921)\) → \((-1,e\left(\frac{5}{16}\right),-1,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 3648 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(1\) | \(e\left(\frac{5}{48}\right)\) |
sage: chi.jacobi_sum(n)