from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,27,25,0]))
pari: [g,chi] = znchar(Mod(101,3240))
Basic properties
Modulus: | \(3240\) | |
Conductor: | \(648\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{648}(101,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3240.cw
\(\chi_{3240}(101,\cdot)\) \(\chi_{3240}(221,\cdot)\) \(\chi_{3240}(461,\cdot)\) \(\chi_{3240}(581,\cdot)\) \(\chi_{3240}(821,\cdot)\) \(\chi_{3240}(941,\cdot)\) \(\chi_{3240}(1181,\cdot)\) \(\chi_{3240}(1301,\cdot)\) \(\chi_{3240}(1541,\cdot)\) \(\chi_{3240}(1661,\cdot)\) \(\chi_{3240}(1901,\cdot)\) \(\chi_{3240}(2021,\cdot)\) \(\chi_{3240}(2261,\cdot)\) \(\chi_{3240}(2381,\cdot)\) \(\chi_{3240}(2621,\cdot)\) \(\chi_{3240}(2741,\cdot)\) \(\chi_{3240}(2981,\cdot)\) \(\chi_{3240}(3101,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((2431,1621,3161,1297)\) → \((1,-1,e\left(\frac{25}{54}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3240 }(101, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{29}{54}\right)\) |
sage: chi.jacobi_sum(n)