from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1155, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,10,54]))
pari: [g,chi] = znchar(Mod(17,1155))
Basic properties
Modulus: | \(1155\) | |
Conductor: | \(1155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1155.dm
\(\chi_{1155}(17,\cdot)\) \(\chi_{1155}(68,\cdot)\) \(\chi_{1155}(173,\cdot)\) \(\chi_{1155}(227,\cdot)\) \(\chi_{1155}(248,\cdot)\) \(\chi_{1155}(332,\cdot)\) \(\chi_{1155}(437,\cdot)\) \(\chi_{1155}(458,\cdot)\) \(\chi_{1155}(563,\cdot)\) \(\chi_{1155}(668,\cdot)\) \(\chi_{1155}(677,\cdot)\) \(\chi_{1155}(887,\cdot)\) \(\chi_{1155}(908,\cdot)\) \(\chi_{1155}(992,\cdot)\) \(\chi_{1155}(1097,\cdot)\) \(\chi_{1155}(1118,\cdot)\)
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
\((386,232,661,211)\) → \((-1,i,e\left(\frac{1}{6}\right),e\left(\frac{9}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 1155 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)