from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7440, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,15,0,29]))
pari: [g,chi] = znchar(Mod(641,7440))
Basic properties
Modulus: | \(7440\) | |
Conductor: | \(93\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{93}(83,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7440.ki
\(\chi_{7440}(641,\cdot)\) \(\chi_{7440}(881,\cdot)\) \(\chi_{7440}(1841,\cdot)\) \(\chi_{7440}(2801,\cdot)\) \(\chi_{7440}(3041,\cdot)\) \(\chi_{7440}(4481,\cdot)\) \(\chi_{7440}(5201,\cdot)\) \(\chi_{7440}(7121,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | \(\Q(\zeta_{93})^+\) |
Values on generators
\((6511,1861,4961,2977,5521)\) → \((1,1,-1,1,e\left(\frac{29}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 7440 }(641, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) |
sage: chi.jacobi_sum(n)