from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7440, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,30,0,46]))
pari: [g,chi] = znchar(Mod(941,7440))
Basic properties
| Modulus: | \(7440\) | |
| Conductor: | \(1488\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1488}(941,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7440.kt
\(\chi_{7440}(941,\cdot)\) \(\chi_{7440}(1181,\cdot)\) \(\chi_{7440}(1541,\cdot)\) \(\chi_{7440}(2501,\cdot)\) \(\chi_{7440}(2621,\cdot)\) \(\chi_{7440}(2741,\cdot)\) \(\chi_{7440}(3341,\cdot)\) \(\chi_{7440}(3701,\cdot)\) \(\chi_{7440}(4661,\cdot)\) \(\chi_{7440}(4901,\cdot)\) \(\chi_{7440}(5261,\cdot)\) \(\chi_{7440}(6221,\cdot)\) \(\chi_{7440}(6341,\cdot)\) \(\chi_{7440}(6461,\cdot)\) \(\chi_{7440}(7061,\cdot)\) \(\chi_{7440}(7421,\cdot)\)
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
\((6511,1861,4961,2977,5521)\) → \((1,-i,-1,1,e\left(\frac{23}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7440 }(941, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) |
sage: chi.jacobi_sum(n)