from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,12,26]))
pari: [g,chi] = znchar(Mod(877,2736))
Basic properties
| Modulus: | \(2736\) | |
| Conductor: | \(2736\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2736.gy
\(\chi_{2736}(205,\cdot)\) \(\chi_{2736}(637,\cdot)\) \(\chi_{2736}(877,\cdot)\) \(\chi_{2736}(925,\cdot)\) \(\chi_{2736}(1021,\cdot)\) \(\chi_{2736}(1237,\cdot)\) \(\chi_{2736}(1573,\cdot)\) \(\chi_{2736}(2005,\cdot)\) \(\chi_{2736}(2245,\cdot)\) \(\chi_{2736}(2293,\cdot)\) \(\chi_{2736}(2389,\cdot)\) \(\chi_{2736}(2605,\cdot)\)
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: | 36.0.1518491026041947944789864372537347836310504032267670560805344803268132977411739903962193307107328.1 |
Values on generators
\((1711,2053,1217,1009)\) → \((1,-i,e\left(\frac{1}{3}\right),e\left(\frac{13}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2736 }(877, a) \) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(-1\) | \(e\left(\frac{5}{36}\right)\) |
sage: chi.jacobi_sum(n)