from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3696, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,30,20,54]))
pari: [g,chi] = znchar(Mod(149,3696))
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.he
\(\chi_{3696}(149,\cdot)\) \(\chi_{3696}(557,\cdot)\) \(\chi_{3696}(821,\cdot)\) \(\chi_{3696}(893,\cdot)\) \(\chi_{3696}(1157,\cdot)\) \(\chi_{3696}(1229,\cdot)\) \(\chi_{3696}(1493,\cdot)\) \(\chi_{3696}(1733,\cdot)\) \(\chi_{3696}(1997,\cdot)\) \(\chi_{3696}(2405,\cdot)\) \(\chi_{3696}(2669,\cdot)\) \(\chi_{3696}(2741,\cdot)\) \(\chi_{3696}(3005,\cdot)\) \(\chi_{3696}(3077,\cdot)\) \(\chi_{3696}(3341,\cdot)\) \(\chi_{3696}(3581,\cdot)\)
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
\((463,2773,2465,1585,673)\) → \((1,i,-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3696 }(149, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)