from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3744, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,9,0,22]))
pari: [g,chi] = znchar(Mod(163,3744))
Basic properties
| Modulus: | \(3744\) | |
| Conductor: | \(416\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{416}(163,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3744.ir
\(\chi_{3744}(163,\cdot)\) \(\chi_{3744}(379,\cdot)\) \(\chi_{3744}(739,\cdot)\) \(\chi_{3744}(955,\cdot)\) \(\chi_{3744}(2035,\cdot)\) \(\chi_{3744}(2251,\cdot)\) \(\chi_{3744}(2611,\cdot)\) \(\chi_{3744}(2827,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.24.31808511574029960248322509834333516654369310400053248.1 |
Values on generators
\((703,2341,2081,2017)\) → \((-1,e\left(\frac{3}{8}\right),1,e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3744 }(163, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) | \(e\left(\frac{19}{24}\right)\) | \(-i\) | \(e\left(\frac{23}{24}\right)\) |
sage: chi.jacobi_sum(n)