from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2025, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([1,0]))
pari: [g,chi] = znchar(Mod(326,2025))
Basic properties
Modulus: | \(2025\) | |
Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{81}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2025.bf
\(\chi_{2025}(101,\cdot)\) \(\chi_{2025}(176,\cdot)\) \(\chi_{2025}(326,\cdot)\) \(\chi_{2025}(401,\cdot)\) \(\chi_{2025}(551,\cdot)\) \(\chi_{2025}(626,\cdot)\) \(\chi_{2025}(776,\cdot)\) \(\chi_{2025}(851,\cdot)\) \(\chi_{2025}(1001,\cdot)\) \(\chi_{2025}(1076,\cdot)\) \(\chi_{2025}(1226,\cdot)\) \(\chi_{2025}(1301,\cdot)\) \(\chi_{2025}(1451,\cdot)\) \(\chi_{2025}(1526,\cdot)\) \(\chi_{2025}(1676,\cdot)\) \(\chi_{2025}(1751,\cdot)\) \(\chi_{2025}(1901,\cdot)\) \(\chi_{2025}(1976,\cdot)\)
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
\((326,1702)\) → \((e\left(\frac{1}{54}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 2025 }(326, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)