from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7098, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,2]))
pari: [g,chi] = znchar(Mod(547,7098))
Basic properties
| Modulus: | \(7098\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{169}(40,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7098.ci
\(\chi_{7098}(547,\cdot)\) \(\chi_{7098}(1093,\cdot)\) \(\chi_{7098}(1639,\cdot)\) \(\chi_{7098}(2185,\cdot)\) \(\chi_{7098}(2731,\cdot)\) \(\chi_{7098}(3277,\cdot)\) \(\chi_{7098}(3823,\cdot)\) \(\chi_{7098}(4369,\cdot)\) \(\chi_{7098}(4915,\cdot)\) \(\chi_{7098}(5461,\cdot)\) \(\chi_{7098}(6007,\cdot)\) \(\chi_{7098}(6553,\cdot)\)
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: | 13.13.542800770374370512771595361.1 |
Values on generators
\((4733,5071,6931)\) → \((1,1,e\left(\frac{1}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7098 }(547, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
sage: chi.jacobi_sum(n)