from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8550, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,3,0]))
pari: [g,chi] = znchar(Mod(77,8550))
Basic properties
Modulus: | \(8550\) | |
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 \(\chi_{225}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8550.ft
\(\chi_{8550}(77,\cdot)\) \(\chi_{8550}(533,\cdot)\) \(\chi_{8550}(1103,\cdot)\) \(\chi_{8550}(1217,\cdot)\) \(\chi_{8550}(1787,\cdot)\) \(\chi_{8550}(2813,\cdot)\) \(\chi_{8550}(2927,\cdot)\) \(\chi_{8550}(3497,\cdot)\) \(\chi_{8550}(3953,\cdot)\) \(\chi_{8550}(4523,\cdot)\) \(\chi_{8550}(4637,\cdot)\) \(\chi_{8550}(5663,\cdot)\) \(\chi_{8550}(6233,\cdot)\) \(\chi_{8550}(6347,\cdot)\) \(\chi_{8550}(6917,\cdot)\) \(\chi_{8550}(7373,\cdot)\)
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
\((1901,1027,1351)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{20}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 8550 }(77, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)