from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([25,0,12]))
pari: [g,chi] = znchar(Mod(401,2475))
Basic properties
Modulus: | \(2475\) | |
Conductor: | \(99\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{99}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2475.eo
\(\chi_{2475}(401,\cdot)\) \(\chi_{2475}(851,\cdot)\) \(\chi_{2475}(1076,\cdot)\) \(\chi_{2475}(1226,\cdot)\) \(\chi_{2475}(1301,\cdot)\) \(\chi_{2475}(1676,\cdot)\) \(\chi_{2475}(1901,\cdot)\) \(\chi_{2475}(2126,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.0.29099190400267368949073680941341991556317731763.1 |
Values on generators
\((551,2377,2026)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{2}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 2475 }(401, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)