from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,10,7]))
pari: [g,chi] = znchar(Mod(4,1911))
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 \(\chi_{637}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1911.dw
\(\chi_{1911}(4,\cdot)\) \(\chi_{1911}(205,\cdot)\) \(\chi_{1911}(277,\cdot)\) \(\chi_{1911}(478,\cdot)\) \(\chi_{1911}(550,\cdot)\) \(\chi_{1911}(751,\cdot)\) \(\chi_{1911}(823,\cdot)\) \(\chi_{1911}(1024,\cdot)\) \(\chi_{1911}(1297,\cdot)\) \(\chi_{1911}(1369,\cdot)\) \(\chi_{1911}(1570,\cdot)\) \(\chi_{1911}(1642,\cdot)\)
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: | 42.42.16423600478713504434070778628293678810006717122913176085381268066336462525553883868883157384200587461557.1 |
Values on generators
\((638,1522,1471)\) → \((1,e\left(\frac{5}{21}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 1911 }(4, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{42}\right)\) |
sage: chi.jacobi_sum(n)