from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,11,28]))
pari: [g,chi] = znchar(Mod(61,1911))
Basic properties
Modulus: | \(1911\) | |
Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{637}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1911.dq
\(\chi_{1911}(61,\cdot)\) \(\chi_{1911}(94,\cdot)\) \(\chi_{1911}(334,\cdot)\) \(\chi_{1911}(367,\cdot)\) \(\chi_{1911}(640,\cdot)\) \(\chi_{1911}(880,\cdot)\) \(\chi_{1911}(1153,\cdot)\) \(\chi_{1911}(1186,\cdot)\) \(\chi_{1911}(1426,\cdot)\) \(\chi_{1911}(1459,\cdot)\) \(\chi_{1911}(1699,\cdot)\) \(\chi_{1911}(1732,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((638,1522,1471)\) → \((1,e\left(\frac{11}{42}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 1911 }(61, a) \) | \(-1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(-1\) | \(e\left(\frac{23}{42}\right)\) |
sage: chi.jacobi_sum(n)