from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1710, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([6,9,16]))
pari: [g,chi] = znchar(Mod(47,1710))
Basic properties
| Modulus: | \(1710\) | |
| Conductor: | \(855\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{855}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1710.dh
\(\chi_{1710}(47,\cdot)\) \(\chi_{1710}(137,\cdot)\) \(\chi_{1710}(347,\cdot)\) \(\chi_{1710}(443,\cdot)\) \(\chi_{1710}(473,\cdot)\) \(\chi_{1710}(833,\cdot)\) \(\chi_{1710}(1073,\cdot)\) \(\chi_{1710}(1127,\cdot)\) \(\chi_{1710}(1157,\cdot)\) \(\chi_{1710}(1163,\cdot)\) \(\chi_{1710}(1373,\cdot)\) \(\chi_{1710}(1517,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.36045670002337036813834863966937246686386512405362460211785986211962997913360595703125.1 |
Values on generators
\((191,1027,1351)\) → \((e\left(\frac{1}{6}\right),i,e\left(\frac{4}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1710 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(1\) | \(i\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) |
sage: chi.jacobi_sum(n)