from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7098, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,13,8]))
pari: [g,chi] = znchar(Mod(1847,7098))
Basic properties
Modulus: | \(7098\) | |
Conductor: | \(3549\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3549}(1847,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7098.ck
\(\chi_{7098}(209,\cdot)\) \(\chi_{7098}(755,\cdot)\) \(\chi_{7098}(1301,\cdot)\) \(\chi_{7098}(1847,\cdot)\) \(\chi_{7098}(2393,\cdot)\) \(\chi_{7098}(2939,\cdot)\) \(\chi_{7098}(3485,\cdot)\) \(\chi_{7098}(4031,\cdot)\) \(\chi_{7098}(4577,\cdot)\) \(\chi_{7098}(5123,\cdot)\) \(\chi_{7098}(5669,\cdot)\) \(\chi_{7098}(6215,\cdot)\)
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
\((4733,5071,6931)\) → \((-1,-1,e\left(\frac{4}{13}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 7098 }(1847, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) |
sage: chi.jacobi_sum(n)