from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4560, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,18,9,8]))
pari: [g,chi] = znchar(Mod(2657,4560))
Basic properties
Modulus: | \(4560\) | |
Conductor: | \(285\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{285}(92,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4560.iq
\(\chi_{4560}(17,\cdot)\) \(\chi_{4560}(593,\cdot)\) \(\chi_{4560}(833,\cdot)\) \(\chi_{4560}(1073,\cdot)\) \(\chi_{4560}(1697,\cdot)\) \(\chi_{4560}(2417,\cdot)\) \(\chi_{4560}(2513,\cdot)\) \(\chi_{4560}(2657,\cdot)\) \(\chi_{4560}(2753,\cdot)\) \(\chi_{4560}(2897,\cdot)\) \(\chi_{4560}(4337,\cdot)\) \(\chi_{4560}(4433,\cdot)\)
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: | Number field defined by a degree 36 polynomial |
Values on generators
\((1711,1141,3041,2737,1921)\) → \((1,1,-1,i,e\left(\frac{2}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 4560 }(2657, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) |
sage: chi.jacobi_sum(n)