from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7600, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,0,9,20]))
pari: [g,chi] = znchar(Mod(207,7600))
Basic properties
| Modulus: | \(7600\) | |
| Conductor: | \(380\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{380}(207,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7600.gp
\(\chi_{7600}(207,\cdot)\) \(\chi_{7600}(2543,\cdot)\) \(\chi_{7600}(2607,\cdot)\) \(\chi_{7600}(2943,\cdot)\) \(\chi_{7600}(3007,\cdot)\) \(\chi_{7600}(3407,\cdot)\) \(\chi_{7600}(4607,\cdot)\) \(\chi_{7600}(5343,\cdot)\) \(\chi_{7600}(5743,\cdot)\) \(\chi_{7600}(6143,\cdot)\) \(\chi_{7600}(7343,\cdot)\) \(\chi_{7600}(7407,\cdot)\)
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
\((4751,5701,5777,401)\) → \((-1,1,i,e\left(\frac{5}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 7600 }(207, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) |
sage: chi.jacobi_sum(n)