from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([1,6]))
pari: [g,chi] = znchar(Mod(602,3025))
Basic properties
Modulus: | \(3025\) | |
Conductor: | \(275\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{275}(52,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3025.bm
\(\chi_{3025}(602,\cdot)\) \(\chi_{3025}(723,\cdot)\) \(\chi_{3025}(887,\cdot)\) \(\chi_{3025}(1183,\cdot)\) \(\chi_{3025}(1322,\cdot)\) \(\chi_{3025}(1613,\cdot)\) \(\chi_{3025}(1667,\cdot)\) \(\chi_{3025}(2653,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((727,2301)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 3025 }(602, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(-i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(-1\) | \(e\left(\frac{9}{20}\right)\) | \(i\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)