from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4725, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,33,30]))
pari: [g,chi] = znchar(Mod(748,4725))
Basic properties
Modulus: | \(4725\) | |
Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1575}(223,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4725.ft
\(\chi_{4725}(748,\cdot)\) \(\chi_{4725}(937,\cdot)\) \(\chi_{4725}(1063,\cdot)\) \(\chi_{4725}(1252,\cdot)\) \(\chi_{4725}(2008,\cdot)\) \(\chi_{4725}(2197,\cdot)\) \(\chi_{4725}(2638,\cdot)\) \(\chi_{4725}(2827,\cdot)\) \(\chi_{4725}(2953,\cdot)\) \(\chi_{4725}(3142,\cdot)\) \(\chi_{4725}(3583,\cdot)\) \(\chi_{4725}(3772,\cdot)\) \(\chi_{4725}(3898,\cdot)\) \(\chi_{4725}(4087,\cdot)\) \(\chi_{4725}(4528,\cdot)\) \(\chi_{4725}(4717,\cdot)\)
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
\((4376,1702,2026)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{11}{20}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 4725 }(748, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) |
sage: chi.jacobi_sum(n)