from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,39,0,32]))
pari: [g,chi] = znchar(Mod(277,3648))
Basic properties
| Modulus: | \(3648\) | |
| Conductor: | \(1216\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1216}(277,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3648.dv
\(\chi_{3648}(277,\cdot)\) \(\chi_{3648}(349,\cdot)\) \(\chi_{3648}(733,\cdot)\) \(\chi_{3648}(805,\cdot)\) \(\chi_{3648}(1189,\cdot)\) \(\chi_{3648}(1261,\cdot)\) \(\chi_{3648}(1645,\cdot)\) \(\chi_{3648}(1717,\cdot)\) \(\chi_{3648}(2101,\cdot)\) \(\chi_{3648}(2173,\cdot)\) \(\chi_{3648}(2557,\cdot)\) \(\chi_{3648}(2629,\cdot)\) \(\chi_{3648}(3013,\cdot)\) \(\chi_{3648}(3085,\cdot)\) \(\chi_{3648}(3469,\cdot)\) \(\chi_{3648}(3541,\cdot)\)
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
\((2623,2053,1217,1921)\) → \((1,e\left(\frac{13}{16}\right),1,e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3648 }(277, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(-1\) | \(e\left(\frac{29}{48}\right)\) |
sage: chi.jacobi_sum(n)