from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,16]))
pari: [g,chi] = znchar(Mod(821,2240))
Basic properties
| Modulus: | \(2240\) | |
| Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{448}(373,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2240.fi
\(\chi_{2240}(221,\cdot)\) \(\chi_{2240}(261,\cdot)\) \(\chi_{2240}(501,\cdot)\) \(\chi_{2240}(541,\cdot)\) \(\chi_{2240}(781,\cdot)\) \(\chi_{2240}(821,\cdot)\) \(\chi_{2240}(1061,\cdot)\) \(\chi_{2240}(1101,\cdot)\) \(\chi_{2240}(1341,\cdot)\) \(\chi_{2240}(1381,\cdot)\) \(\chi_{2240}(1621,\cdot)\) \(\chi_{2240}(1661,\cdot)\) \(\chi_{2240}(1901,\cdot)\) \(\chi_{2240}(1941,\cdot)\) \(\chi_{2240}(2181,\cdot)\) \(\chi_{2240}(2221,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((1471,1541,897,1921)\) → \((1,e\left(\frac{5}{16}\right),1,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 2240 }(821, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)