from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(374, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([24,5]))
pari: [g,chi] = znchar(Mod(9,374))
Basic properties
Modulus: | \(374\) | |
Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{187}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 374.r
\(\chi_{374}(9,\cdot)\) \(\chi_{374}(15,\cdot)\) \(\chi_{374}(25,\cdot)\) \(\chi_{374}(49,\cdot)\) \(\chi_{374}(53,\cdot)\) \(\chi_{374}(59,\cdot)\) \(\chi_{374}(93,\cdot)\) \(\chi_{374}(179,\cdot)\) \(\chi_{374}(185,\cdot)\) \(\chi_{374}(213,\cdot)\) \(\chi_{374}(223,\cdot)\) \(\chi_{374}(229,\cdot)\) \(\chi_{374}(247,\cdot)\) \(\chi_{374}(257,\cdot)\) \(\chi_{374}(291,\cdot)\) \(\chi_{374}(355,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | 40.40.24562817038400928776197921239227357886542077974183334844678041435576602047153.1 |
Values on generators
\((35,309)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) | \(25\) |
\( \chi_{ 374 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)