from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9405, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,45,48,0]))
pari: [g,chi] = znchar(Mod(58,9405))
Basic properties
| Modulus: | \(9405\) | |
| Conductor: | \(495\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{495}(58,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9405.lc
\(\chi_{9405}(58,\cdot)\) \(\chi_{9405}(742,\cdot)\) \(\chi_{9405}(1312,\cdot)\) \(\chi_{9405}(1483,\cdot)\) \(\chi_{9405}(2623,\cdot)\) \(\chi_{9405}(3193,\cdot)\) \(\chi_{9405}(3877,\cdot)\) \(\chi_{9405}(4162,\cdot)\) \(\chi_{9405}(5758,\cdot)\) \(\chi_{9405}(5872,\cdot)\) \(\chi_{9405}(6043,\cdot)\) \(\chi_{9405}(7297,\cdot)\) \(\chi_{9405}(7582,\cdot)\) \(\chi_{9405}(7753,\cdot)\) \(\chi_{9405}(9007,\cdot)\) \(\chi_{9405}(9178,\cdot)\)
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
\((1046,1882,5986,496)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{4}{5}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(23\) | \(26\) |
| \( \chi_{ 9405 }(58, a) \) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{5}\right)\) |
sage: chi.jacobi_sum(n)