from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4029, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,2]))
pari: [g,chi] = znchar(Mod(3178,4029))
Basic properties
Modulus: | \(4029\) | |
Conductor: | \(1343\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1343}(492,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4029.bn
\(\chi_{4029}(67,\cdot)\) \(\chi_{4029}(220,\cdot)\) \(\chi_{4029}(526,\cdot)\) \(\chi_{4029}(934,\cdot)\) \(\chi_{4029}(1444,\cdot)\) \(\chi_{4029}(2566,\cdot)\) \(\chi_{4029}(2617,\cdot)\) \(\chi_{4029}(3127,\cdot)\) \(\chi_{4029}(3178,\cdot)\) \(\chi_{4029}(3382,\cdot)\) \(\chi_{4029}(3484,\cdot)\) \(\chi_{4029}(3892,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
Fixed field: | Number field defined by a degree 26 polynomial |
Values on generators
\((2687,3556,3163)\) → \((1,-1,e\left(\frac{1}{13}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 4029 }(3178, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) |
sage: chi.jacobi_sum(n)