from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,0,26]))
pari: [g,chi] = znchar(Mod(79,3648))
Basic properties
| Modulus: | \(3648\) | |
| 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 \(\chi_{304}(155,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3648.dl
\(\chi_{3648}(79,\cdot)\) \(\chi_{3648}(751,\cdot)\) \(\chi_{3648}(1039,\cdot)\) \(\chi_{3648}(1135,\cdot)\) \(\chi_{3648}(1231,\cdot)\) \(\chi_{3648}(1807,\cdot)\) \(\chi_{3648}(1903,\cdot)\) \(\chi_{3648}(2575,\cdot)\) \(\chi_{3648}(2863,\cdot)\) \(\chi_{3648}(2959,\cdot)\) \(\chi_{3648}(3055,\cdot)\) \(\chi_{3648}(3631,\cdot)\)
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
\((2623,2053,1217,1921)\) → \((-1,i,1,e\left(\frac{13}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3648 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{36}\right)\) |
sage: chi.jacobi_sum(n)