from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7440, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,0,52]))
pari: [g,chi] = znchar(Mod(421,7440))
Basic properties
| Modulus: | \(7440\) | |
| Conductor: | \(496\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{496}(421,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7440.ks
\(\chi_{7440}(421,\cdot)\) \(\chi_{7440}(541,\cdot)\) \(\chi_{7440}(661,\cdot)\) \(\chi_{7440}(1621,\cdot)\) \(\chi_{7440}(1981,\cdot)\) \(\chi_{7440}(2221,\cdot)\) \(\chi_{7440}(3181,\cdot)\) \(\chi_{7440}(3541,\cdot)\) \(\chi_{7440}(4141,\cdot)\) \(\chi_{7440}(4261,\cdot)\) \(\chi_{7440}(4381,\cdot)\) \(\chi_{7440}(5341,\cdot)\) \(\chi_{7440}(5701,\cdot)\) \(\chi_{7440}(5941,\cdot)\) \(\chi_{7440}(6901,\cdot)\) \(\chi_{7440}(7261,\cdot)\)
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
\((6511,1861,4961,2977,5521)\) → \((1,i,1,1,e\left(\frac{13}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7440 }(421, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) |
sage: chi.jacobi_sum(n)