from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3696, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,0,50,54]))
pari: [g,chi] = znchar(Mod(61,3696))
Basic properties
Modulus: | \(3696\) | |
Conductor: | \(1232\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1232}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3696.hg
\(\chi_{3696}(61,\cdot)\) \(\chi_{3696}(325,\cdot)\) \(\chi_{3696}(733,\cdot)\) \(\chi_{3696}(997,\cdot)\) \(\chi_{3696}(1069,\cdot)\) \(\chi_{3696}(1333,\cdot)\) \(\chi_{3696}(1405,\cdot)\) \(\chi_{3696}(1669,\cdot)\) \(\chi_{3696}(1909,\cdot)\) \(\chi_{3696}(2173,\cdot)\) \(\chi_{3696}(2581,\cdot)\) \(\chi_{3696}(2845,\cdot)\) \(\chi_{3696}(2917,\cdot)\) \(\chi_{3696}(3181,\cdot)\) \(\chi_{3696}(3253,\cdot)\) \(\chi_{3696}(3517,\cdot)\)
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
\((463,2773,2465,1585,673)\) → \((1,-i,1,e\left(\frac{5}{6}\right),e\left(\frac{9}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3696 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)