from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3696, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,0,10,3]))
pari: [g,chi] = znchar(Mod(79,3696))
Basic properties
| Modulus: | \(3696\) | |
| Conductor: | \(308\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{308}(79,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3696.gp
\(\chi_{3696}(79,\cdot)\) \(\chi_{3696}(415,\cdot)\) \(\chi_{3696}(655,\cdot)\) \(\chi_{3696}(1327,\cdot)\) \(\chi_{3696}(1663,\cdot)\) \(\chi_{3696}(1999,\cdot)\) \(\chi_{3696}(2767,\cdot)\) \(\chi_{3696}(3439,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.1123215710861738275456915198522447563874269573052104704.1 |
Values on generators
\((463,2773,2465,1585,673)\) → \((-1,1,1,e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3696 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)