from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6025, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([12,9]))
pari: [g,chi] = znchar(Mod(839,6025))
Basic properties
Modulus: | \(6025\) | |
Conductor: | \(6025\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6025.ez
\(\chi_{6025}(79,\cdot)\) \(\chi_{6025}(509,\cdot)\) \(\chi_{6025}(839,\cdot)\) \(\chi_{6025}(1089,\cdot)\) \(\chi_{6025}(1144,\cdot)\) \(\chi_{6025}(1494,\cdot)\) \(\chi_{6025}(2844,\cdot)\) \(\chi_{6025}(2939,\cdot)\) \(\chi_{6025}(3054,\cdot)\) \(\chi_{6025}(3194,\cdot)\) \(\chi_{6025}(3379,\cdot)\) \(\chi_{6025}(5014,\cdot)\) \(\chi_{6025}(5034,\cdot)\) \(\chi_{6025}(5584,\cdot)\) \(\chi_{6025}(5779,\cdot)\) \(\chi_{6025}(5984,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((2652,2176)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{9}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 6025 }(839, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) |
sage: chi.jacobi_sum(n)