from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3696, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,30,10,42]))
pari: [g,chi] = znchar(Mod(227,3696))
Basic properties
| Modulus: | \(3696\) | |
| Conductor: | \(3696\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3696.ha
\(\chi_{3696}(227,\cdot)\) \(\chi_{3696}(299,\cdot)\) \(\chi_{3696}(563,\cdot)\) \(\chi_{3696}(635,\cdot)\) \(\chi_{3696}(899,\cdot)\) \(\chi_{3696}(1139,\cdot)\) \(\chi_{3696}(1403,\cdot)\) \(\chi_{3696}(1811,\cdot)\) \(\chi_{3696}(2075,\cdot)\) \(\chi_{3696}(2147,\cdot)\) \(\chi_{3696}(2411,\cdot)\) \(\chi_{3696}(2483,\cdot)\) \(\chi_{3696}(2747,\cdot)\) \(\chi_{3696}(2987,\cdot)\) \(\chi_{3696}(3251,\cdot)\) \(\chi_{3696}(3659,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((463,2773,2465,1585,673)\) → \((-1,-i,-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3696 }(227, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) |
sage: chi.jacobi_sum(n)