from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,54,40]))
pari: [g,chi] = znchar(Mod(1018,1045))
Basic properties
| Modulus: | \(1045\) | |
| Conductor: | \(1045\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 1045.cf
\(\chi_{1045}(7,\cdot)\) \(\chi_{1045}(68,\cdot)\) \(\chi_{1045}(83,\cdot)\) \(\chi_{1045}(178,\cdot)\) \(\chi_{1045}(182,\cdot)\) \(\chi_{1045}(277,\cdot)\) \(\chi_{1045}(292,\cdot)\) \(\chi_{1045}(387,\cdot)\) \(\chi_{1045}(448,\cdot)\) \(\chi_{1045}(558,\cdot)\) \(\chi_{1045}(657,\cdot)\) \(\chi_{1045}(733,\cdot)\) \(\chi_{1045}(767,\cdot)\) \(\chi_{1045}(843,\cdot)\) \(\chi_{1045}(942,\cdot)\) \(\chi_{1045}(1018,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((837,761,496)\) → \((-i,e\left(\frac{9}{10}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(1018, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(-i\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) |
sage: chi.jacobi_sum(n)