from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,33,10]))
pari: [g,chi] = znchar(Mod(73,1575))
Basic properties
| Modulus: | \(1575\) | |
| Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{175}(73,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1575.ei
\(\chi_{1575}(73,\cdot)\) \(\chi_{1575}(208,\cdot)\) \(\chi_{1575}(262,\cdot)\) \(\chi_{1575}(388,\cdot)\) \(\chi_{1575}(397,\cdot)\) \(\chi_{1575}(523,\cdot)\) \(\chi_{1575}(577,\cdot)\) \(\chi_{1575}(703,\cdot)\) \(\chi_{1575}(712,\cdot)\) \(\chi_{1575}(838,\cdot)\) \(\chi_{1575}(892,\cdot)\) \(\chi_{1575}(1027,\cdot)\) \(\chi_{1575}(1153,\cdot)\) \(\chi_{1575}(1333,\cdot)\) \(\chi_{1575}(1342,\cdot)\) \(\chi_{1575}(1522,\cdot)\)
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
\((1226,127,451)\) → \((1,e\left(\frac{11}{20}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 1575 }(73, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) |
sage: chi.jacobi_sum(n)