from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9450, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,51,40]))
pari: [g,chi] = znchar(Mod(8747,9450))
Basic properties
Modulus: | \(9450\) | |
Conductor: | \(525\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{525}(347,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9450.fq
\(\chi_{9450}(53,\cdot)\) \(\chi_{9450}(863,\cdot)\) \(\chi_{9450}(1187,\cdot)\) \(\chi_{9450}(1997,\cdot)\) \(\chi_{9450}(2753,\cdot)\) \(\chi_{9450}(3077,\cdot)\) \(\chi_{9450}(3833,\cdot)\) \(\chi_{9450}(3887,\cdot)\) \(\chi_{9450}(4967,\cdot)\) \(\chi_{9450}(5723,\cdot)\) \(\chi_{9450}(5777,\cdot)\) \(\chi_{9450}(6533,\cdot)\) \(\chi_{9450}(7613,\cdot)\) \(\chi_{9450}(7667,\cdot)\) \(\chi_{9450}(8423,\cdot)\) \(\chi_{9450}(8747,\cdot)\)
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
\((9101,6427,6751)\) → \((-1,e\left(\frac{17}{20}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 9450 }(8747, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)