from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5445, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,15,6]))
pari: [g,chi] = znchar(Mod(112,5445))
Basic properties
| Modulus: | \(5445\) | |
| Conductor: | \(495\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{495}(112,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5445.cg
\(\chi_{5445}(112,\cdot)\) \(\chi_{5445}(403,\cdot)\) \(\chi_{5445}(457,\cdot)\) \(\chi_{5445}(1183,\cdot)\) \(\chi_{5445}(1492,\cdot)\) \(\chi_{5445}(1933,\cdot)\) \(\chi_{5445}(2218,\cdot)\) \(\chi_{5445}(2272,\cdot)\) \(\chi_{5445}(2653,\cdot)\) \(\chi_{5445}(3022,\cdot)\) \(\chi_{5445}(3307,\cdot)\) \(\chi_{5445}(3742,\cdot)\) \(\chi_{5445}(3748,\cdot)\) \(\chi_{5445}(4468,\cdot)\) \(\chi_{5445}(4813,\cdot)\) \(\chi_{5445}(4837,\cdot)\)
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
\((3026,4357,3511)\) → \((e\left(\frac{1}{3}\right),i,e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 5445 }(112, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)