from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9450, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,39,0]))
pari: [g,chi] = znchar(Mod(5167,9450))
Basic properties
Modulus: | \(9450\) | |
Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{225}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9450.ga
\(\chi_{9450}(127,\cdot)\) \(\chi_{9450}(253,\cdot)\) \(\chi_{9450}(883,\cdot)\) \(\chi_{9450}(1387,\cdot)\) \(\chi_{9450}(2017,\cdot)\) \(\chi_{9450}(2773,\cdot)\) \(\chi_{9450}(3277,\cdot)\) \(\chi_{9450}(4033,\cdot)\) \(\chi_{9450}(4663,\cdot)\) \(\chi_{9450}(5167,\cdot)\) \(\chi_{9450}(5797,\cdot)\) \(\chi_{9450}(5923,\cdot)\) \(\chi_{9450}(6553,\cdot)\) \(\chi_{9450}(7687,\cdot)\) \(\chi_{9450}(7813,\cdot)\) \(\chi_{9450}(8947,\cdot)\)
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
\((9101,6427,6751)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{13}{20}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 9450 }(5167, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)