from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9600, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,1,16,0]))
pari: [g,chi] = znchar(Mod(251,9600))
Basic properties
| Modulus: | \(9600\) | |
| Conductor: | \(384\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{384}(251,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9600.em
\(\chi_{9600}(251,\cdot)\) \(\chi_{9600}(851,\cdot)\) \(\chi_{9600}(1451,\cdot)\) \(\chi_{9600}(2051,\cdot)\) \(\chi_{9600}(2651,\cdot)\) \(\chi_{9600}(3251,\cdot)\) \(\chi_{9600}(3851,\cdot)\) \(\chi_{9600}(4451,\cdot)\) \(\chi_{9600}(5051,\cdot)\) \(\chi_{9600}(5651,\cdot)\) \(\chi_{9600}(6251,\cdot)\) \(\chi_{9600}(6851,\cdot)\) \(\chi_{9600}(7451,\cdot)\) \(\chi_{9600}(8051,\cdot)\) \(\chi_{9600}(8651,\cdot)\) \(\chi_{9600}(9251,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.32.135104323545903136978453058557785670637514001130337144105502507008.1 |
Values on generators
\((4351,901,6401,5377)\) → \((-1,e\left(\frac{1}{32}\right),-1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 9600 }(251, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(-i\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)