from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7803, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([28,27]))
pari: [g,chi] = znchar(Mod(616,7803))
Basic properties
| Modulus: | \(7803\) | |
| Conductor: | \(459\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{459}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7803.bb
\(\chi_{7803}(616,\cdot)\) \(\chi_{7803}(1483,\cdot)\) \(\chi_{7803}(1696,\cdot)\) \(\chi_{7803}(2563,\cdot)\) \(\chi_{7803}(3217,\cdot)\) \(\chi_{7803}(4084,\cdot)\) \(\chi_{7803}(4297,\cdot)\) \(\chi_{7803}(5164,\cdot)\) \(\chi_{7803}(5818,\cdot)\) \(\chi_{7803}(6685,\cdot)\) \(\chi_{7803}(6898,\cdot)\) \(\chi_{7803}(7765,\cdot)\)
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
\((2891,2026)\) → \((e\left(\frac{7}{9}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 7803 }(616, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) |
sage: chi.jacobi_sum(n)