from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,15,0,40]))
pari: [g,chi] = znchar(Mod(971,2240))
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 \(\chi_{448}(75,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2240.fh
\(\chi_{2240}(131,\cdot)\) \(\chi_{2240}(171,\cdot)\) \(\chi_{2240}(411,\cdot)\) \(\chi_{2240}(451,\cdot)\) \(\chi_{2240}(691,\cdot)\) \(\chi_{2240}(731,\cdot)\) \(\chi_{2240}(971,\cdot)\) \(\chi_{2240}(1011,\cdot)\) \(\chi_{2240}(1251,\cdot)\) \(\chi_{2240}(1291,\cdot)\) \(\chi_{2240}(1531,\cdot)\) \(\chi_{2240}(1571,\cdot)\) \(\chi_{2240}(1811,\cdot)\) \(\chi_{2240}(1851,\cdot)\) \(\chi_{2240}(2091,\cdot)\) \(\chi_{2240}(2131,\cdot)\)
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
\((1471,1541,897,1921)\) → \((-1,e\left(\frac{5}{16}\right),1,e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 2240 }(971, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)