from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,45,10,24]))
pari: [g,chi] = znchar(Mod(4273,6930))
Basic properties
Modulus: | \(6930\) | |
Conductor: | \(3465\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3465}(808,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6930.iv
\(\chi_{6930}(103,\cdot)\) \(\chi_{6930}(367,\cdot)\) \(\chi_{6930}(493,\cdot)\) \(\chi_{6930}(1237,\cdot)\) \(\chi_{6930}(1753,\cdot)\) \(\chi_{6930}(2623,\cdot)\) \(\chi_{6930}(2887,\cdot)\) \(\chi_{6930}(3127,\cdot)\) \(\chi_{6930}(4273,\cdot)\) \(\chi_{6930}(4387,\cdot)\) \(\chi_{6930}(4513,\cdot)\) \(\chi_{6930}(4777,\cdot)\) \(\chi_{6930}(5647,\cdot)\) \(\chi_{6930}(5773,\cdot)\) \(\chi_{6930}(6037,\cdot)\) \(\chi_{6930}(6163,\cdot)\)
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
\((1541,1387,2971,2521)\) → \((e\left(\frac{2}{3}\right),-i,e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 6930 }(4273, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)