from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2925, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,9,40]))
pari: [g,chi] = znchar(Mod(308,2925))
Basic properties
| Modulus: | \(2925\) | |
| Conductor: | \(2925\) | 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 2925.gh
\(\chi_{2925}(302,\cdot)\) \(\chi_{2925}(308,\cdot)\) \(\chi_{2925}(542,\cdot)\) \(\chi_{2925}(653,\cdot)\) \(\chi_{2925}(887,\cdot)\) \(\chi_{2925}(1127,\cdot)\) \(\chi_{2925}(1238,\cdot)\) \(\chi_{2925}(1472,\cdot)\) \(\chi_{2925}(1478,\cdot)\) \(\chi_{2925}(1712,\cdot)\) \(\chi_{2925}(1823,\cdot)\) \(\chi_{2925}(2063,\cdot)\) \(\chi_{2925}(2297,\cdot)\) \(\chi_{2925}(2408,\cdot)\) \(\chi_{2925}(2642,\cdot)\) \(\chi_{2925}(2648,\cdot)\)
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
\((326,352,2251)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 2925 }(308, a) \) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(-i\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) |
sage: chi.jacobi_sum(n)