from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7098, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,9]))
pari: [g,chi] = znchar(Mod(1429,7098))
Basic properties
| Modulus: | \(7098\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{169}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7098.co
\(\chi_{7098}(883,\cdot)\) \(\chi_{7098}(1429,\cdot)\) \(\chi_{7098}(1975,\cdot)\) \(\chi_{7098}(2521,\cdot)\) \(\chi_{7098}(3067,\cdot)\) \(\chi_{7098}(3613,\cdot)\) \(\chi_{7098}(4159,\cdot)\) \(\chi_{7098}(4705,\cdot)\) \(\chi_{7098}(5251,\cdot)\) \(\chi_{7098}(5797,\cdot)\) \(\chi_{7098}(6343,\cdot)\) \(\chi_{7098}(6889,\cdot)\)
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: | 26.26.3830224792147131369362629348887201408953937846517364173.1 |
Values on generators
\((4733,5071,6931)\) → \((1,1,e\left(\frac{9}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7098 }(1429, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) |
sage: chi.jacobi_sum(n)