from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2025, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([32,27]))
pari: [g,chi] = znchar(Mod(49,2025))
Basic properties
Modulus: | \(2025\) | |
Conductor: | \(405\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{405}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2025.be
\(\chi_{2025}(49,\cdot)\) \(\chi_{2025}(124,\cdot)\) \(\chi_{2025}(274,\cdot)\) \(\chi_{2025}(349,\cdot)\) \(\chi_{2025}(499,\cdot)\) \(\chi_{2025}(574,\cdot)\) \(\chi_{2025}(724,\cdot)\) \(\chi_{2025}(799,\cdot)\) \(\chi_{2025}(949,\cdot)\) \(\chi_{2025}(1024,\cdot)\) \(\chi_{2025}(1174,\cdot)\) \(\chi_{2025}(1249,\cdot)\) \(\chi_{2025}(1399,\cdot)\) \(\chi_{2025}(1474,\cdot)\) \(\chi_{2025}(1624,\cdot)\) \(\chi_{2025}(1699,\cdot)\) \(\chi_{2025}(1849,\cdot)\) \(\chi_{2025}(1924,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((326,1702)\) → \((e\left(\frac{16}{27}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 2025 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) |
sage: chi.jacobi_sum(n)