from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,28]))
pari: [g,chi] = znchar(Mod(913,1728))
Basic properties
Modulus: | \(1728\) | |
Conductor: | \(432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{432}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1728.bs
\(\chi_{1728}(49,\cdot)\) \(\chi_{1728}(241,\cdot)\) \(\chi_{1728}(337,\cdot)\) \(\chi_{1728}(529,\cdot)\) \(\chi_{1728}(625,\cdot)\) \(\chi_{1728}(817,\cdot)\) \(\chi_{1728}(913,\cdot)\) \(\chi_{1728}(1105,\cdot)\) \(\chi_{1728}(1201,\cdot)\) \(\chi_{1728}(1393,\cdot)\) \(\chi_{1728}(1489,\cdot)\) \(\chi_{1728}(1681,\cdot)\)
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: | 36.36.614667125325361522818798575155151578949632894783197825857500612833312768.1 |
Values on generators
\((703,325,1217)\) → \((1,-i,e\left(\frac{7}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 1728 }(913, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)