from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7600, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,0,10]))
pari: [g,chi] = znchar(Mod(51,7600))
Basic properties
| Modulus: | \(7600\) | |
| Conductor: | \(304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{304}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7600.fx
\(\chi_{7600}(51,\cdot)\) \(\chi_{7600}(451,\cdot)\) \(\chi_{7600}(851,\cdot)\) \(\chi_{7600}(2651,\cdot)\) \(\chi_{7600}(3251,\cdot)\) \(\chi_{7600}(3651,\cdot)\) \(\chi_{7600}(3851,\cdot)\) \(\chi_{7600}(4251,\cdot)\) \(\chi_{7600}(4651,\cdot)\) \(\chi_{7600}(6451,\cdot)\) \(\chi_{7600}(7051,\cdot)\) \(\chi_{7600}(7451,\cdot)\)
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: | 36.36.19036714782161565107424425435655777110146017378670996611401194085493506048.1 |
Values on generators
\((4751,5701,5777,401)\) → \((-1,-i,1,e\left(\frac{5}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 7600 }(51, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{35}{36}\right)\) |
sage: chi.jacobi_sum(n)