from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2541, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,11,9]))
pari: [g,chi] = znchar(Mod(703,2541))
Basic properties
| Modulus: | \(2541\) | |
| Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{847}(703,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2541.br
\(\chi_{2541}(10,\cdot)\) \(\chi_{2541}(208,\cdot)\) \(\chi_{2541}(439,\cdot)\) \(\chi_{2541}(472,\cdot)\) \(\chi_{2541}(670,\cdot)\) \(\chi_{2541}(703,\cdot)\) \(\chi_{2541}(901,\cdot)\) \(\chi_{2541}(934,\cdot)\) \(\chi_{2541}(1132,\cdot)\) \(\chi_{2541}(1165,\cdot)\) \(\chi_{2541}(1363,\cdot)\) \(\chi_{2541}(1396,\cdot)\) \(\chi_{2541}(1594,\cdot)\) \(\chi_{2541}(1627,\cdot)\) \(\chi_{2541}(1825,\cdot)\) \(\chi_{2541}(1858,\cdot)\) \(\chi_{2541}(2089,\cdot)\) \(\chi_{2541}(2287,\cdot)\) \(\chi_{2541}(2320,\cdot)\) \(\chi_{2541}(2518,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((848,1816,2059)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 2541 }(703, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)