from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7440, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,0,15,56]))
pari: [g,chi] = znchar(Mod(7,7440))
Basic properties
| Modulus: | \(7440\) | |
| Conductor: | \(1240\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1240}(627,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7440.lt
\(\chi_{7440}(7,\cdot)\) \(\chi_{7440}(103,\cdot)\) \(\chi_{7440}(727,\cdot)\) \(\chi_{7440}(1063,\cdot)\) \(\chi_{7440}(2167,\cdot)\) \(\chi_{7440}(2407,\cdot)\) \(\chi_{7440}(2983,\cdot)\) \(\chi_{7440}(3367,\cdot)\) \(\chi_{7440}(3703,\cdot)\) \(\chi_{7440}(4327,\cdot)\) \(\chi_{7440}(4567,\cdot)\) \(\chi_{7440}(5143,\cdot)\) \(\chi_{7440}(5383,\cdot)\) \(\chi_{7440}(5527,\cdot)\) \(\chi_{7440}(6343,\cdot)\) \(\chi_{7440}(7303,\cdot)\)
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
\((6511,1861,4961,2977,5521)\) → \((-1,-1,1,i,e\left(\frac{14}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7440 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) |
sage: chi.jacobi_sum(n)