from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2541, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,44,54]))
pari: [g,chi] = znchar(Mod(1222,2541))
Basic properties
Modulus: | \(2541\) | |
Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{847}(375,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2541.bo
\(\chi_{2541}(67,\cdot)\) \(\chi_{2541}(100,\cdot)\) \(\chi_{2541}(298,\cdot)\) \(\chi_{2541}(331,\cdot)\) \(\chi_{2541}(529,\cdot)\) \(\chi_{2541}(562,\cdot)\) \(\chi_{2541}(760,\cdot)\) \(\chi_{2541}(793,\cdot)\) \(\chi_{2541}(991,\cdot)\) \(\chi_{2541}(1024,\cdot)\) \(\chi_{2541}(1222,\cdot)\) \(\chi_{2541}(1255,\cdot)\) \(\chi_{2541}(1486,\cdot)\) \(\chi_{2541}(1684,\cdot)\) \(\chi_{2541}(1717,\cdot)\) \(\chi_{2541}(1915,\cdot)\) \(\chi_{2541}(1948,\cdot)\) \(\chi_{2541}(2146,\cdot)\) \(\chi_{2541}(2377,\cdot)\) \(\chi_{2541}(2410,\cdot)\)
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 33 polynomial |
Values on generators
\((848,1816,2059)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{9}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 2541 }(1222, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) |
sage: chi.jacobi_sum(n)