from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(53312, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,1,0]))
pari: [g,chi] = znchar(Mod(10881,53312))
Basic properties
| Modulus: | \(53312\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 53312.me
\(\chi_{53312}(6529,\cdot)\) \(\chi_{53312}(10881,\cdot)\) \(\chi_{53312}(14145,\cdot)\) \(\chi_{53312}(18497,\cdot)\) \(\chi_{53312}(21761,\cdot)\) \(\chi_{53312}(26113,\cdot)\) \(\chi_{53312}(29377,\cdot)\) \(\chi_{53312}(33729,\cdot)\) \(\chi_{53312}(36993,\cdot)\) \(\chi_{53312}(41345,\cdot)\) \(\chi_{53312}(48961,\cdot)\) \(\chi_{53312}(52225,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((51647,3333,10881,40769)\) → \((1,1,e\left(\frac{1}{42}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 53312 }(10881, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) |
sage: chi.jacobi_sum(n)