from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,10,18]))
pari: [g,chi] = znchar(Mod(277,6045))
Basic properties
| Modulus: | \(6045\) | |
| Conductor: | \(2015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2015}(277,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6045.nz
\(\chi_{6045}(277,\cdot)\) \(\chi_{6045}(1453,\cdot)\) \(\chi_{6045}(2038,\cdot)\) \(\chi_{6045}(2383,\cdot)\) \(\chi_{6045}(2662,\cdot)\) \(\chi_{6045}(2968,\cdot)\) \(\chi_{6045}(3208,\cdot)\) \(\chi_{6045}(3247,\cdot)\) \(\chi_{6045}(3592,\cdot)\) \(\chi_{6045}(4138,\cdot)\) \(\chi_{6045}(4177,\cdot)\) \(\chi_{6045}(4183,\cdot)\) \(\chi_{6045}(4417,\cdot)\) \(\chi_{6045}(5113,\cdot)\) \(\chi_{6045}(5347,\cdot)\) \(\chi_{6045}(5392,\cdot)\)
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
\((4031,4837,1861,2731)\) → \((1,i,e\left(\frac{1}{6}\right),e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 6045 }(277, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) |
sage: chi.jacobi_sum(n)